Multiplicity-1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature

We address the one-parameter minmax construction, via Allen--Cahn energy, that has recently lead to a new proof of the existence of a closed minimal hypersurface in an arbitrary closed Riemannian manifold $N^{n+1}$ with $n\geq 2$ (see Guaraco's 2018 work). We obtain the following multiplicity-$1$ result: if the Ricci curvature of $N$ is positive then the minmax Allen--Cahn solutions concentrate around a multiplicity-$1$ hypersurface, that may have a singular set of dimension $\leq n-7$. This result is new for $n\geq 3$ (for $n=2$ it is also implied by recent work by Chodosh--Mantoulidis). The argument developed here is geometric in flavour and exploits directly the minmax characterization of the solutions. As immediate corollary we obtain that every closed Riemannian manifold $N^{n+1}$ with $n\geq 2$ and positive Ricci curvature admits a two-sided closed minimal hypersurface, possibly with a singular set of dimension $\leq n-7$. This existence result is new for $n\geq 7$ (for $2\leq n \leq 6$, where no singular set is present, it also follows from multiplicity-$1$ results developed within the Almgren--Pitts framework, see works by Ketover-Marques-Neves, Zhou, Marques-Neves).


Introduction
The existence of a closed minimal hypersurface in an arbitrary closed Riemannian manifold N n+1 (n ≥ 2) is a fundamental result that dates back to the early 80s: the minmax procedure developed by Almgren and Pitts [2] [19] leads to a stationary integral varifold whose support is, by the regularity/compactness theory of Schoen-Simon-Yau [21] and Schoen-Simon [22], a smoothly embedded hypersurface except possibly for a singular set of codimension 7 or higher.This existence result has been recently reproven, using an alternative minmax approach based on an Allen-Cahn approximation scheme (summarized briefly in Section 2.1), through the combined efforts of Guaraco, Hutchinson, Tonegawa, Wickramasekera [10], [12], [24], [25], [27].As in the former approach, one obtains the minimal hypersurface as an integral varifold, that turns out to be smooth away from a singular set of codimension ≥ 7 (moreover, its regular part has Morse index at most one by [11], [7]).In general, in both approaches, the hypersurface may have several connected components, each a priori with an integer multiplicity ≥ 1.
In the Almgren-Pitts framework, Ketover-Marques-Neves [15] show that, when N n+1 is orientable with positive Ricci curvature and 2 ≤ n ≤ 6, the minimal hypersurface is two-sided and has multiplicity 1.More recently, Zhou [28] obtains multiplicity-1 and two-sidedness in the case 2 ≤ n ≤ 6 for bumpy metrics and for metrics with positive Ricci, more generally for one-or multi-parameter minmax, thereby confirming a well-known conjecture of Marques-Neves [17, 1.2] (see also [17,Addendum]).
Marques-Neves's conjecture has a natural counterpart for minmax constructions via Allen-Cahn.For n = 2 it follows from Chodosh-Mantoulidis [5] (valid more generally for solutions with bounded Morse index, not necessarily minmax solutions) that the minimal surface obtained by the (one-or multi-parameter) Allen-Cahn minmax is twosided and has multiplicity 1, in the case of bumpy metrics and in the case of metrics with positive Ricci.
We obtain here the following multiplicity-1 result (new for n ≥ 3), which applies to the Allen-Cahn one-parameter minmax construction in [10].
Theorem 1.1.Let N be a closed Riemannian manifold of dimension n + 1 with n ≥ 2 and with positive Ricci curvature.Then the Allen-Cahn minmax in [10] yields on N a multiplicity-1 smooth minimal hypersurface M with dim M \ M ≤ n − 7.
The multiplicity issue is quite ubiquitous in variational problems and its resolution has far-reaching consequences, specifically (but not only) in minmax constructions, see e.g.[5], [17], [28] and, in yet another minmax approach (for surfaces in arbitrary codimension), [18].In the case of Theorem 1.1 we have, implicitly in the multiplicity-1 conclusion, the following existence result for two-sided minimal hypersurfaces.
Theorem 1.2.In any closed Riemannian manifold of dimension n + 1 with n ≥ 2 and with positive Ricci curvature there exists a smooth two-sided minimal hypersurface M with dim M \ M ≤ n − 7.
Theorem 1.2 is new for n ≥ 7, where the presence of a potential singular set has so far hindered the use of many techniques developed in the low-dimensional setting over the past years.
Remark 1.1.It also follows easily that M in Theorem 1.1 is connected.

Strategy
To obtain the multiplicity-1 result we exploit directly the minmax characterization (rather than finite index properties).More precisely, we do not analyse the Allen-Cahn solutions constructed in [10], that concentrate on the minimal hypersurface; we only retain the minmax value that they achieve and prove the following result, from which Theorem 1.1 is easily deduced.Theorem 1.3.Let N be a closed Riemannian manifold of dimension n + 1, n ≥ 2, and with positive Ricci curvature.Let M ⊂ N be any smooth minimal hypersurface such that dim M \ M ≤ n − 7, M is stationary in N , and for every x ∈ M there exists a geodesic ball in N centred at x in which M is stable.Then the minmax value c ε obtained by [10] (for ε < 1) satisfies Remark 1.2.The assumptions on M in Theorem 1.3 are valid for any minimal hypersurface produced by the minmax in [10] (using [12], [24], [25], [27]).Then it is readily checked that Theorem 1.1 follows from Theorem 1.3.
Remark 1.3.It is not hard to check that, under the assumptions of Theorem 1.1, the area of the minmax hypersurface is less than or equal to that of an arbitrary two-sided minimal hypersurface in N that has the properties listed for M in Theorem 1.3.
Remark 1.4.Part of the arguments developed here does not make use of the Ricci positive assumption.In fact, analogues of Theorems 1.1 and 1.2 in the case in which 2 ≤ n ≤ 6 and N is endowed with a bumpy metric will appear in [4].
We will now outline the proof of Theorem 1.3.Given M as in Theorem 1.3, the idea is to produce, for all sufficiently small ε, a continuous path in W 1,2 (N ) that joins the constant −1 to the constant +1 and such that the Allen-Cahn energy evaluated along the path stays below 2H n (M ) by a fixed positive amount independent of ε (determined only by geometric properties of M ⊂ N ).Since this is one of the admissible paths for the minmax construction in [10], the inequality in Theorem 1.3 must hold.
The construction of the path is geometric in flavour.For simplicity, in this introduction we illustrate it mainly in the case 2 ≤ n ≤ 6, so that M is smooth and closed.We think of M with multiplicity 2 as an immersed two-sided hypersurface, namely its double cover M with the standard projection.This immersion, that we denote by ι : M → N , is minimal and unstable (by the positiveness of the Ricci curvature).It is possible to find a (sufficiently small) geodesic ball B ⊂ M such that the lack of stability still holds for deformations that do not move B (this follows by a capacity argument).We then find a deformation of ι that is area-decreasing on some time interval [0, t 0 ].This deformation is depicted in the top row of Figure 1.(We can choose the initial speed of the deformation to be non-negative on M , therefore the deformation "pushes away from M ").We denote by 2H n (M )−τ the area of the immersion at time t 0 , for some τ > 0. If we cut out B from M we are left with an immersion with boundary, namely ι| M \ι −1 (B) .We can restrict the previous deformation to ι| M \ι −1 (B) , obtaining an area-decreasing deformation (at fixed boundary) on the time interval [0, t 0 ].This time the area changes from 2H n (M ) − 2H n (B) to 2H n (M ) − 2H n (B) − τ .This deformation is depicted in middle row of Figure 1.Now we proceed to close the hole at B continuously (bottom row of Figure 1), reaching, say in in time 1, the same immersion depicted in the top right picture of Figure 1.The area changes from 2H n (M ) − 2H n (B) − τ to 2H n (M ) − τ (staying below the final value).Therefore, in going from the middle-left picture to the bottom-right picture of Figure 1, we have produced a "path of immersions" along which the value of the area stays stricly below 2H n (M ), at least by min{τ, 2H n (B)}, a fixed amount that only depends on the geometry of M ⊂ N .
This path of immersions is then "reproduced at the Allen-Cahn level", i.e. replaced by a continuous path γ : [0, t 0 + 1] → W 1,2 (N ).Each function in the image if this curve is a suitable "Allen-Cahn approximation" of the corresponding immersion: to do this, one fits one-dimensional Allen-Cahn solutions in the normal bundle to the immersion, respecting multiplicities.(At points with multiplicity 1 and 2 we will use respectively the top and bottom profiles in Figure 3; the operation of closing the hole at B can be reproduced at the Allen-Cahn level because we have multiplicity-2 on B, moreover the operation is continuous in W 1,2 (N ).)This construction is done for all sufficiently small ε (the parameter of the Allen-Cahn energy) and, moreover, for all sufficiently small ε the Allen-Cahn energy all along γ is a close approximation of the area of the corresponding immersions; therefore, for all sufficiently small ε, the energies stay below 2H n (M ) by a fixed "geometric" amount ≈ min{τ, 2H n (B)}.
We now consider γ(0) and γ(t 0 + 1) (respectively the Allen-Cahn approximations of the immersions in the middle-left and top-right picture of Figure 1).For the latter, we use a (negative) Allen-Cahn gradient flow (to which we add a small forcing term).We build a mean convex barrier (by writing a suitable Allen-Cahn approximation of ι), that controls γ(t 0 +1) from below.This forces the flow to deform γ(t 0 +1) continuously to a stable Allen-Cahn solution, which has to be the constant +1 by the Ricci-positive assumption.Along this flow, the Allen-Cahn energy is controlled by the initial bound ≈ 2H n (M ) − τ .The function γ(0) is ≈ +1 close to M \ B and ≈ −1 away from a tube around M \ B: this function can be explicitly connected to the constant −1 (continuously in W 1,2 ) with approximately decreasing Allen-Cahn energy.Reversing the latter path, then composing it with γ and finally with the path obtained via the flow, we produce the promised continuous path in W 1,2 (N ) that joins −1 to +1 and has the desired energy control.We stress that the functions γ(t), t ∈ [0, t 0 + 1], that we call "Allen-Cahn approximations" of the corresponding immersions, are not solutions of the Allen-Cahn equation, they only realize the "correct" energy value.In fact, for t = t 0 + 1, we do not even analyse the Allen-Cahn first variation of γ(t).The loss of information on the first variation is compensated by the ad hoc structure of the Allen-Cahn approximation: its level sets are by construction sheets over the given immersed hypersurface, so that the Allen-Cahn energy is an effective approximation of area (by the coarea formula) and the geometric information can be translated to the Allen-Cahn level.
We emphasise the following point of view on the idea described above.Consider ι : M → N : we exhibit two one-sided deformations that decrease area and that can be reproduced for the Allen-Cahn approximations.One (from the top-left to the middleleft picture of Figure 1) is given by the operation of piercing a hole at the centre of B and enlarging the hole until 2|B| is removed (replicating this for the Allen-Cahn approximations uses the multiplicity-2 assumption).The other (from the top-left to the top-right picture of Figure 1) is a deformation of ι as an immersion, induced by an initial velocity compactly supported away from B. The two deformations are linearly independent, since the first only acts on B and the latter only moves the hypersurface outside B. We will denote by G ε 0 in Section 4 the Allen-Cahn approximation of ι.Then, with rerefence to Figure 2, and using the notation γ(0), γ(t 0 + 1) respectively for the Figure 2: Lowering the peak (landscape for the Allen-Cahn energy).The same labels as in Figure 1 are used, to denote deformations that reproduce those in Figure 1.
Allen-Cahn approximations of the immersions in the middle-left and top-right picture of Figure 1, the two deformations just described, implemented at the Allen-Cahn level, correspond respectively to "going from G ε 0 to γ(0)" and "going from G ε 0 to γ(t 0 + 1)".Note that it may well be that ι is an immersion with Morse index 1 (for example, the double cover of an equator of RP 3 ).The area-decreasing deformation obtained by piercing a hole at the centre of B and enlarging it until all of B has been removed is clearly not a deformation of ι as an immersion; it can be reproduced for Allen-Cahn approximations thanks to the fact that we have multiplicity 2 on B, so that the profile in the normal bundle to B looks like the bottom one in Figure 3 and can be connected continuously to the constant −1 with controlled energy.
The function γ(0) (that will be denoted by f = g 0 in Section 7) can be connected to the constant −1 and the function γ(t 0 + 1) can be connected to the constant +1, as described in the sketch.We thus have a "recovery path" for the value 2H n (M ): this path connects −1 to +1 and the maximum on the path is ≈ 2H n (M ).What we achieve is to deform this path in the sorroundings of G ε 0 , exploiting the information that we have gained on the landscape, specifically we deform the portion between γ(0) and γ(t 0 + 1).From γ(0) we use a deformation that reproduces the one in the middle row of Figure 1.By doing this we reach the function g t 0 (notation from Section 7).Now we close the hole continuously, replicating the deformation in the bottom row of Figure 1, reaching the function γ(t 0 + 1) (that will be denoted by g t 0 +1 in Section 7).We have thus found a path γ : [0, t 0 + 1] → W 1,2 (N ), from γ(0) to γ(t 0 + 1), that lowers the peak, compared to the initial "recovery path".This follows thanks to the fact that the Allen-Cahn energy is a close approximation of the area of the corresponding immersion, so we inherit the estimates that we had for the path of immersions that joins the middle-left picture to the bottom-right picture of Figure 1.
The argument shows that the landscape around G ε 0 is reminiscent of one where the Morse index is ≥ 2. However G ε 0 is not a stationary point for the Allen-Cahn energy.In fact, we never need to compute the Allen-Cahn second variation along the deformations, it suffices to know that the Allen-Cahn energy has strictly decreased at γ(0) and γ(t 0 + 1), compared to its value at G ε 0 .We digress to comment briefly on the operation of connecting γ(0) to the constant −1.We could in fact use an Allen-Cahn flow for this step, by first slightly deforming γ(0) into another function (with a similar profile, so that it still approximates 2|M \ B|, but with a more effective first variation) and then running the Allen-Cahn flow, that deforms this function to the constant −1.We do not argue in this way, since we are able to produce an explicit deformation of γ(0) to −1, and this is more straightforward.We point out, however, that the deformation that we exhibit mimics exactly the Allen-Cahn flow, and is therefore a regularized version of the Brakke flow that starts at 2|M \ B| and vanishes instantaneously.While the Brakke flow creates a discontinuity in space-time, at the Allen-Cahn level we gain continuity (and the flow reaches It is conceivable that the portion of the path from γ(0) to +1 can also be imitated (e.g. in an Almgren-Pitts framework) by a one-parameter family of boundaries in N , by doubling M \ B and inserting a neck at B, then pushing this hypersurface away from M without moving the neck (and decreasing the area), then closing the neck, then using mean-curvature-flow to drift away from M until extinction time.However, for the portion of path that goes from the Allen-Cahn approximation of γ(0) to the constant −1, it seems more challenging to construct a corresponding one-parameter family of boundaries in N .The Allen-Cahn framework allows a very straightforward solution to this issue.
For n ≥ 7, we still employ the idea illustrated above.Its implementation, however, is rendered somewhat harder by the presence of the singular set: standard tubular neighbourhoods and Fermi coordinates for M (that are essential to fit one-dimensional Allen-Cahn profiles in the normal bundle to M ) are not available.While the geometric ideas remain the same as in the low-dimensional case, we need to additionally study certain analytic properties.Denote by d M : N → [0, ∞) the distance function to M .We observe that, if d M (x) is smaller than the injectivity radius of N , then d M (x) is always realized by a geodesic (possibly more than one) from x to a smooth point of M .This allows to analyse the cut-locus of d M (restricting to {d M < inj(N )}), following [16], and obtain n-rectifiability properties for it.This leads to the existence of a suitable replacement for Fermi coordinates, which becomes the usual one on any compact subset of M .Denote by ι : M → N the immersion given by the standard projection from the double cover of M .We choose K ⊂ M compact (sufficiently large) and a geodesic ball B ⊂ ι(K) (sufficiently small) so that ι : M → N admits a deformation as an immersion that decreases area and only moves K \ ι −1 (B).(This is analogous to what we did in the lower dimensional case, except that this time we additionally need a deformation that does not move M close to the singular set.)The set K will play the role that was of M in the low-dimensional case.Around ι(K) we define Allen-Cahn approximations of suitable immersions by fitting one-dimensional Allen-Cahn profiles in the normal bundle.Away from ι(K), we use the level sets of d M to complete the definition of the desired Allen-Cahn approximations and create (as in the low-dimensional case) a continuous path γ : [0, t 0 +1] → W 1,2 (N ).(This appears to be simpler than obtaining an analogous continuous family of singular hypersurfaces, in fact we only prove that almost every level set of d M is almost everywhere smooth; our Allen-Cahn approximations are, close to the singular set, built from level sets of d M and an almost everywhere control is sufficient to ensure that they realize the "correct" energy value.)Moreover, exploiting the n-rectifiability of the cut-locus and analysing the singular part of the measure ∆d M (and using the Ricci-positive condition) we obtain that (restricting to {d M < inj(N )}) the (distributional) Laplacian of d M is a positive Radon measure.This translates into a mean convexity property for the Allen-Cahn approximation G ε 0 of ι : M → N , thanks to which we obtain a mean-convex smooth barrier m for the negative Allen-Cahn gradient flow (we add a small constant forcing term).By employing m we produce the part of the path that connects γ(t 0 + 1) to the constant +1.(In this step as well there appears to be a gain in simplicity by working with Allen-Cahn approximations rather than directly with singular hypersurfaces.)

Remarks for n ≤ 6
After the preliminary Section 2, we begin the proof, which we write for n ≥ 7, assuming the existence of a singular set M \ M of dimension ≤ n − 7.While the underlying ideas are the same for all dimensions, the proof becomes considerably shorter and more straightforward in the absence of singular set, in particular when n ≤ 6.In detail, Sections 3 and 4, in which we study the distance function to M and its level sets, can be omitted when M = M and one can use standard facts about tubular neighbourhoods of smooth closed hypersurfaces.The compact set K that we need to work with in Sections 5 and 6 can be replaced simply by M when M = M , and therefore the definitions of the Allen-Cahn approximations of the relevant immersions given in Section 7 become simpler.Finally, in Section 7.5 we construct a barrier m by suitably mollifying a Lipschitz function G ε 0 , which is defined from the level sets of d M and is an Allen-Cahn approximation of ι : M → N .This convolution procedure (described in Appendix A) ensures smoothness and mean-convexity of m, which is technically important for our arguments.If M = M , G ε 0 is already smooth and mean-convex, therefore no smoothing is needed and Appendix A and a portion of Section 7.5 can be omitted.
Acknowledgments: This work is partially supported by EPSRC under grant EP/S005641/1.I am thankful to Neshan Wickramasekera for introducing me to the Allen-Cahn functional and its geometric impact.I would like to thank Felix Schulze for helpful conversations about parabolic PDEs and mean curvature flow.I am grateful to Otis Chodosh for a mini-course on geometric features of the Allen-Cahn equation, held at Princeton University in June 2019, and for related discussions.The insight that I gained turned out to be very valuable when I addressed the problem discussed in this work.These lectures took place while I was a member of the Institute for Advanced Study, Princeton: I gratefully acknowledge the excellent research environment and the support provided by the Institute and by the National Science Foundation under Grant No. DMS-1638352.

Preliminaries
We give a brief summmary of the construction in [10], then introduce the onedimensional Allen-Cahn profiles that will be needed for our proof.

Reminders: Allen-Cahn minmax approximation scheme
We recall the minmax construction in [10].For ε ∈ (0, 1) consider the functional Here W is a C 2 double well potential with nondegenerate minima at ±1, for example , suitably modified (for technical reasons) outside [−2, 2]-a typical choice, that we also follow, is to impose quadratic growth to ∞ (others are also possible), and σ is a normalization constant, σ = 1 −1 W (t)/2 dt.Consider continuous paths in W 1,2 (N ) that start at the constant −1 and end at the constant +1: this is the class of admissible paths.A "wall condition" is ensured and yields the existence of a minmax solution u ε to E ε (u ε ) = 0.Moreover, upper and lower energy bounds are established, uniformly in ε.
In order to produce a stationary varifold, one considers w ε = Φ(u ε ) as in [12] (with The analysis in [12] (which only requires the stationarity of u ε and no assumption on their second variation), together with the upper and lower bounds for E ε (u ε ), gives that V ε converges subsequentially, as ε → 0, to an integral n-varifold V = 0 with vanishing first variation.Thanks to the fact that the Morse index of u ε is ≤ 1 for all ε, [10] reduces the problem locally in N to one that concerns stable Allen-Cahn solutions, as in [24].For these, the regularity theory of [27] and [25] applies and gives that spt V is smoothly embedded away from a possible singular set of dimension ≤ n−7, i.e.V is the varifold of integration over a finite set of closed minimal hypersurfaces, each counted with integer multiplicity: V = K j=1 q j |M j |, with q j ∈ N and M j minimal and smooth away from a set of dimension ≤ n − 7 (|M j | denotes the multiplicity-1 varifold of integration on M j ).In the case n ≤ 6 all the M j 's are completely smooth.(In the case Ric N > 0 there is only one connected component, K = 1, see Remark 8.1.) We point out that, denoting by ε i the sequence extracted to guarantee the varifold convergence, E ε i (u ε i ) → V (N ) in this construction, in other words the Allen-Cahn energy of u ε i converges to the mass K j=1 q j H n (M j ) of V .

1-dimensional profiles
Let H(r) denote the monotonically increasing solution to u − W (u) = 0 such that lim r→±∞ H(r) = ±1, with H(0) = 0. (For the standard potential we have ) Then also H(−r) and H(±r + z) solve u − W (u) = 0 (for any z ∈ R).The rescaled function Truncations.The arguments developed here will involve the construction of suitable Allen-Cahn approximations of certain immersions.For that purpose, we will make use of approximate versions of H ε .While this introcuces small errors in the corresponding ODEs, it has the advantage that the approximate solutions are constant (±1) away from an interval of the form An Allen-Cahn approximation of a hypersurface in N requires to fit the 1-dimensional profiles in the normal direction to the hypersurface and we need to stay inside a tubular neighbourhood, so it is effective to have one-dimensional profiles that become constant before we reach the boundary of the tubular neighbourhood.

Computation of the Allen-Cahn energy of H
ε .To compute the energy of H ε , following [13], we have, for any q : R → R, The first term vanishes when q = H ε .Let G denote a primitive of 2W (t).For the second term, noting that the integrand is G(q) , we get G(q(b))−G(q(a)).In particular, Recalling the definition of H ε , we have that Families of profiles.Define the function Ψ : R → R This function is smooth thanks to the fact that all derivatives of H ε vanish at ±2 ε Λ.
Define the following evolution for t ∈ [0, ∞): Note that Ψ 0 = Ψ and The functions Ψ t form a family of even, Lipschitz functions and E ε (Ψ t ) is decreasing in t.Indeed, the energy contribution of the "tails" (±1) is zero (so the energy is finite) and we have ε .The profiles Ψ t , and profiles of the type H ε (• − t), will be used within our construction to produce Allen-Cahn approximations of relevant immersions (possibly with boundary), having multiplicity 1 or 2 on their image.

Distance function to M
Let N be a Riemannian manifold of dimension n + 1 with n ≥ 2 and with positive Ricci curvature Ric N > 0. Let M ⊂ N be a smooth minimal hypersurface with dim M \ M ≤ n − 7, M is stationary in N , and M is locally stable in N , i.e. for every point in M there exists a geodesic ball centred at the point in which M is stable.These properties are true for the ε → 0 varifold limit of finite-index Allen-Cahn solutions on N , thanks to the analysis in [10], [12], [24], [25], [27].The stationarity condition implies the existence of tangent cones at every point in M .A consequence of the deep sheeting theorem in [22], [27] is that any point of M at which one tangent cone is supported on a hyperplane has to be a smooth point.
Let dist N denote the (unsigned) Riemannian distance on N ; we will be interested in the function Since N is complete, for every x the value d M (x) is realized by at least one geodesic from x to M (Hopf-Rinow).We recall a few facts that are true of the distance to an arbitrary closed set, see [16, Section 3]1 .The function d M is Lipschitz on N (with constant 1) and locally semi-concave on N \ M , so that its gradient is BV loc on N \ M (equivalently, the distributional Hessian of d M on N \ M is a Radon measure).We denote by S d M the subset of N \ M where d M fails to be differentiable; S d M coincides with the set of points in p ∈ N \ M for which there exist at least two geodesics from p to M whose length realizes and S d M is countably n-rectifiable (this uses [1]).However, rectifiability is not necessarily true of its closure unless extra hypotheses on the closed set are available; for example, S d M would be countably n-rectifiable (even in the C k−2 sense) if M were a C k submanifold with k ≥ 3, thanks to [16,Section 4].While this statement does not apply immediately in our case due to the presence of the singular set M \ M , the proof in [16,Section 4] can still be carried out without any change by virtue of the following key observation.Lemma 3.1.Let x ∈ N \ M .For any geodesic from x to M (whose length realizes d M (x)), we have that the endpoint y on M actually belongs to M .
Proof.Let γ be any geodesic from x to M , let y be its endpoint on M and fix a point z ∈ N that lies in the image of γ and such that dist N (z, y) < inj(N ).Consider the (open) geodesic ball B(z) ⊂ N centred at z with radius dist N (z, y).Then M ∩B(z) = ∅ (otherwise there would be a shorter curve than γ joining x to M ) and y ∈ M ∩ ∂B(z).Since the monotonicity formula holds at all points of M (M is stationary in N ), we can blow up at y to obtain tangent cones.Then every tangent cone to M at y has to be supported in a (closed) half space (the complement of the open half space obtained by blowing up B(z) at y).By [23, Ch. 7 Theorem 4.5, Remark 4.6] every tangent cone to M at y is the hyperplane tangent to B(z) at y, possibly with multiplicity.As pointed out above, the sheeting theorem in [22], [27] implies that y is a smooth point.
In other words, any geodesic that realizes the distance to M has to end at a smooth point, i.e. on M (and it meets M orthogonally).This allows to repeat the arguments in [16,Section 4] (as we briefly sketch below) and obtain Proposition 3.1.In the rest of this work we will be interested in the set T ω = {x ∈ N : dist N (x, M ) < ω}, where ω is chosen in (0, inj(N )), therefore we restrict to this open set for our analysis (even though not strictly necessary for this section).Proposition 3.1 (as in [16]).The set S d M ∩ T ω is countably n-rectifiable2 .Moreover, ∇d M ∈ SBV loc (T ω \ M ) and the singular part (with respect to Remark 3.1.Additionally we have, since M is smooth, that the absolutely continuous part of D 2 d M has a smooth density with respect to . This density coincides with the pointwise Hessian of d M . The rectifiability of S d M implies that at H n -a.e. point x ∈ S d M there exists a measure-theoretic unit normal nx to S d M (rather, two choices of it).The rectifiability of S d M and the smoothness of d M on T ω \ (M ∪ S d M ) imply that the "Cantor part" of the BV loc function D 2 d M is 0 (see the sketch below), leaving only the "jump part", which is supported on S d M .This means that ∇d M is SBV loc (T ω \ M ); moreover, for H n -a.e. point x ∈ S d M ∩ T ω , the left and right limits in the Lebesgue sense of ∇d M at x are well-defined in the two halfspaces identified by the normal (see [3,Theorem 3.77]).
Sketch: relevant arguments in [16].Consider the map F (y, v, t) = exp y (tv) for y ∈ M and v a unit vector orthogonal to M at y.For fixed (y, v) the curve F (y, v, t) is a geodesic leaving M orthogonally.We will limit ourselves to t ≤ ω, since we are only interested in T ω .If t 0 is sufficiently small (depending on (y, v)) the geodesic t ∈ [0, t 0 ] → F (y, v, t) is the minimizing curve between its endpoint F (y, v, t 0 ) and M , equivalently, its length is d M (F (y, v, t 0 )).However for large enough t 0 the geodesic may fail to be minimizing, therefore one can consider σ = σ y,v ∈ (0, ω] defined as follows: σ = ω if F (y, v, t) is minimizing (between its endpoint and M ) for all t ≤ ω; otherwise, σ is chosen in (0, ω) so that F (y, v, t) is minimizing (between its endpoint and M ) for t ≤ σ and F (y, v, t) is not minimizing if t > σ.The set of points is the restriction to T ω of the so-called cut-locus of M , and it is a subset of T ω \ M whose closure in T ω does not intersect M .Recall that the unit sphere bundle of M is just M , the oriented double cover of M , so we will also write (y, v) ∈ M .
Using these facts, the arguments of [16,Proposition 4.8] adapt to give that therefore in order to prove the rectifiability in Proposition 3.1 it suffices (since e. the analogue of [16,Theorem 4.11].Note that M \M is H n -negligeable, so it does not affect rectifiability.The points in Cut(M )\S d M are characterised by the validity of option (b) above, and the arguments in [16] are local around the points (y, v, σ y,v ) ∈ M × (0, ω), so they apply verbatim to our case.
Once the countable n-rectifiability of S d M has been obtained, it follows that ∇d M is SBV loc (T ω \ M ).Indeed, we know to begin with (see above for these statements about the distance to a closed set) that ∇d M is in BV loc (T ω \ M ) and notice that d M is C 2 (even C k for all k) on T ω \ (S d M ∪ M ) thanks to the smoothness of M .The "Cantor part" of the Radon measure D 2 d M gives 0 measure to countably n-rectifiable sets (see [3,Prop. 3.92 or Prop.4.2]), in particular it gives This concludes the sketch of proof of Proposition 3.1.
Remark 3.2 (on the diffeomorphism F ).We point out a couple of further facts, mainly adapted from [16], for future reference.The level sets of d M are smooth in the open set T ω \ (S d M ∪ M ), thanks to the implicit function theorem, the smoothness of d M and the invertibility of F on this open set.
The map F (y, v, t) = exp y (tv) for y ∈ M and v a unit vector orthogonal to M at y, t ∈ (0, ω) is a map from M × (0, ω) into T ω (since the oriented double cover M of M is defined as the set of (y, v) with y ∈ M , v unit vector normal to M at y). Arguing as in [16,Prop. 4.8] we see that the following restriction of F (still denoted by F ) is a (smooth) diffeomorphism. 3This diffeomorphism extends as a continuous map to M × {0} by sending ((y, v), 0) to y ∈ M (note that it is a 2 − 1 map here, the standard projection from M to M ).The image of this continuous map is then Again following verbatim [16,Prop. 4.8], we also have that the function σ (y,v) is continuous on M .The diffeomorpshism F in (4), continuously extended to M , provides the natural replacements for Fermi coordinates around M in our situation, where the singular set M \ M is present.We will write for the domain of (the extension of) F .Let us take a closer look at the level sets Γ t = {x ∈ T ω : d M (x) = t}, for t ∈ (0, ω).By the previous discussion, the smooth hypersurface Γ t \ S d M can be restracted smoothly, staying in T ω \ S d M onto a subset of M and at each time the image of the retraction is contained in (a smooth portion of) a level set of d M .In fact, we have a retraction explicitly given, using the identification (4), by (here q = (y, v) ∈ M and σ q = σ (y,v) ) Under the identification (4), the function s is just d M , so it follows that the retraction preserves level sets of d M .
We will now analyse the jump part of the Hessian of d M : T ω \ M → (0, ∞); this will lead to Lemma 3.2 below.To this end, we perform, for H n -a.e. point x ∈ S d M , a blow up of d M as follows.Using normal coordinates around x, for all sufficiently small ρ > 0 consider the function Then (∇d ρ )(y) = (∇d M )(x+ρy).Note that d ρ have Lipschitz-constant 1 and d ρ (0) = 0, therefore we can extract a sequence ρ j → 0 such that d ρ j converge in C 0,α (for all α < 1) to a 1-Lipschitz function Recall that for H n -a.e. point x ∈ S d M we have a normal nx and moreover ∇d M admits one-sided limits in L 1 : there exists two constant vectors a = b in R n+1 such that , where we used, in the two limits, respectively the uniform convergence d ρ j → d x and the L 1 -convergence ∇d ρ j → F ab .The equality obtained expresses the fact that F ab = ∇d x and proves that Recall now that d M is locally semiconcave, so it has at least an element in the superdifferential, i.e. there exists a C 1 function in a neighbourhood of x that is ≥ d M and such that (x) = d M (x).Performing the same blow up on , we consider the rescalings (x+ρy)− (x) ρ .These functions converge in C 1 (B 1 (0)) to an affine function The  [14], [27].Lemma 3.2 is sufficient for our scopes.
Next we analyse the absolutely continuous part (with respect to H n+1 ) of ∆d M , for d M : T ω \ M → (0, ∞).By Proposition 3.1 it suffices to analyse the smooth function ∆d M on T ω \ (S d M ∪ M ).For this, we will need the Ricci positive condition (which has not been used so far).
and its scalar mean curvature at x (with respect to the normal that points away from M ) is −∆d M (x).We are thus in the classical situation in which we look at level sets of the distance function to a smooth submanifold, in this case a geodesic ball B r (π(x)) in M .This gives the information on the Laplacian in a neighbourhood of x.By Riccati's equation [8,Corollary 3.6] we get that in positive Ricci the mean curvature of the level sets {y In conclusion, from Lemmas 3.2 and 3.3, we have ∆d M ≤ 0 on T ω \ M in the sense of distributions.Since d M is smooth at points in M and M is minimal, we also get that ∆d M is 0 on M and so we can extend the previous conclusion: ∆d M ≤ 0 on T ω \ (M \ M ).We will now extend across M \ M by a capacity argument.Proof.Let δ > 0 be arbitrary and choose χ ∈ C ∞ c (T ω ) to be a function that takes values in [0, 1], is identically 1 in an open neighbourhood of M \ M , identically 0 away from a (larger) neighbourhood of M \ M and such that Tω |∇χ| < δ (see [6, 4.7]).Then we have, for For the second term recall that the distribution ∇d M is an a.e. and so This tends to 0 as δ → 0.
As δ → 0, the corresponding χ will go to 0 in L 1 so the third term will also tend to 0.
The distribution ∆d M is a priori of order ≤ 1: For the first term in the right-most side of (6), observe that 0 by the choice of χ).We argue as follows: (6) holds for all δ and its last two terms tend to 0 as δ → 0, therefore for every v ∈ C ∞ c (T ω ) and v ≥ 0 we have The distribution ∆d M is therefore a negative Radon measure on T ω .

Level sets of d M
We consider the level sets Γ t = {x : d M (x) = t}, for t ∈ [0, ω/2] (we fixed an arbitrary ω ∈ (0, inj(N ))); we will obtain that the areas of Γ t are "essentially" decreasing in t.Further, we will consider an "Allen-Cahn approximation" defined, for ε sufficiently small (to ensure 4 ε Λ < ω/2), as follows: Since in T ω and 0 otherwise) is in BV (N ) and its distributional Laplacian ∆G ε 0 is a Radon measure (see Section 3).Note that the profile of G ε 0 in the normal direction at any point of M is given by the function Ψ = Ψ 0 in (3), therefore G ε 0 can also be thought of as an Allen-Cahn approximation of 2|M |, or equivalently of the immersion ι : M → N that covers M twice.The fact that E ε (G ε 0 ) is approximately 2|M | will be etablished later.
The Allen-Cahn first variation of G ε 0 (which is clearly 0 outside T ω ) can be computed in T ω as follows: in the distributional sense.Since ∆d M a Radon measure thanks to Proposition 3.2, we will think of −E ε (G ε 0 ) as a Radon measure.(The term O(ε 2 ) in the last line is a Lipschitz function that we interpret as a density with respect to H n+1 ; the last term is the measure ∆d M multiplied by a bounded Lipschitz function.)Denote by F ε,µ , for a constant µ > 0, the functional on W 1,2 (N ) given by The computation in (8) shows that for every ε there exists (The inequality means that the Radon measure on the left minus the Radon measure on the right is a non-negative measure.)The function G ε 0 will form the starting point for the construction of a barrier for the negative F ε,µ ε -gradient flow in Section 7.5.
Areas of Γ t .Since S d M is countably n-rectifiable (and thus has Hausdorff dimension ≤ n and vanishing H n+1 measure) we get that, for a.e.t > 0, H n (S d M ∩ Γ t ) = 0. We will denote by Ω ⊂ (0, ω) the set with (and therefore, for t / ∈ Ω, Γ t is a smooth hypersurface away from a H n -negligeable set).Therefore for t ∈ (0, ω) \ Ω we have H n (Γ t ) = H n (Γ t \ S d M ), i.e. we only need to compute the area of the smooth part of Γ t .Thanks to this, we will compare the area of Γ t to that of M for t ∈ (0, ω) \ Ω.
Proof.The first part of (a) has already been discussed above.Recall the diffeomorphism induced by F in Remark 3.2.Endow {(q, s) : q ∈ M , s ∈ [0, σ q )} with the pull-back metric (via F ) from T ω \ Cut(M ) \ M .The metric extend continuously to M × {0} to give the natural metric on M .We will thus work in V M = {(q, s) : q ∈ M , s ∈ [0, σ q )}; Denoting by Π the map Π(q, s) = (q, 0), recall that from the structure of V M we obtain the following.For every t < t 0 the Let (x 1 , . . ., x n , s) be local coordinates on V M chosen so that ∂ ∂x 1 , . . ., ∂ ∂xn form a local frame around a point x 0 ∈ M , that is orthonormal at x 0 ∈ M , and ∂ ∂s is the unit speed of the geodesics {x = const}.Then the Riemannian metric on V M induces an area element θ s 0 for the level set {s = s 0 } at the point (x 0 , s 0 ).By [8,Theorem 3.11] it satisfies the ODE ∂ ∂s log θ s = − H(x 0 , s) • ∂ ∂s , where H (x 0 ,s) is the mean curvature of the level set at distance s evaluated at the point (x 0 , s). (Note that in [8] θ s denotes the volume element, but since ∂ ∂s is a unit vector, the area and volume elements are the same.)By Riccati's equation [8,Corollary 3.6] we find that H(x 0 , s) = H (x 0 ,s) • ∂ ∂s is strictly increasing in s, at least at linear rate, thanks to the positiveness of the Ricci curvature, H (x 0 ,s) ≥ s(min N Ric N ).Therefore ∂ ∂s log θ s ≤ −s(min N Ric N ) and we find s ds and therefore In particular, θ(t) is decreasing in t.From this (a) and (b) follow by integrating the area element.(Recall that M θ 0 dx 1 . . .dx n = 2H n (M ).) Allen-Cahn energy of G ε 0 .Thanks to Lemma 4.1 we can control the Allen-Cahn energy of G ε 0 by twice the area of M .Indeed, recalling that the energy is 0 in the complement of T ω/2 and that ∇G ε 0 is parallel to ∇d M , we use the coarea formula for the slicing function d M (for which |∇d M | = 1) and we get where we used Lemma 4.1 (a) for a.e.s, namely s / ∈ Ω.By the estimates in (1) we get 5 Instability properties of M (choice of B) Let ι : M → N be the standard projection (2-1 map) from the oriented double cover of M onto M .This is a smooth minimal immersion.Let ν be a choice (on M ) of unit normal to the immersion ι.Recall (Remark 3.2) the coordinates ((y, v), s) = (q, s) on V M , which is diffeomorphic to T ω \ S d M \ (M \ M ); here y ∈ M and v a unit vector orthogonal to M at y, or, equivalently, q = (y, v) ∈ M .For every compact set K ⊂ M there exists c K > 0 such that c K < σ (y,v) for all (y, v) ∈ K.This follows from the continuity of σ q on M (Remark 3.2).Choosing K even (i.e.such that K is the double cover of a compact set ι(K) in M ) this means that ι(K) admits a two-sided tubular neighbourhood of semi-width c K .
We will now consider deformations of ι with initial velocity dictated by a function ϕ ∈ C 2 c ( M ).For ϕ ∈ C 2 c ( M ), choose c suppϕ as above and consider the following one-parameter family of immersions ι t : M → N defined for t ∈ (−δ 0 , δ 0 ), where δ 0 ∈ 0, csuppϕ maxϕ : (y, v) → exp ι(y) (tϕ((y, v))ν((y, v))), for (y, v) ∈ M .The first variation of area at t = 0 is 0 because M is minimal.The second variation of area at t = 0 is given by where A denotes the second fundamental form of ι, ∇ the gradient on M (with respect to g 0 , the Riemannian metric induced by the pull-back from M ), Ric N the Ricci tensor of N and H n is induced on M by g 0 (equivalently, integrate with repsect to dvol g 0 ).

Lemma 5.1 (unstable region).
There exist a geodesic ball Proof.The second variation of M is only defined for initial velocities induced by a function with compact support in M .Fix an arbitrary point b ∈ M .Let δ > 0 be arbitrary and choose Then the function ϕ(q) = 1 − ρ(ι(q)) is admissible in (10) and the expression becomes (integrating on M ) (Note that on M the choice of ν is in general only permitted up to sign; this suffices for the term Ric N (ν, ν) to make sense.)Sending δ → 0 the second term tends to −2 M (|A M | 2 +Ric N (ν, ν))dH n and the first term tends to 0 (recall that M |A M | 2 < ∞ thanks to the stability assumption) so the above expression converges to a negative number (because Ric N > 0).Therefore there exists δ sufficiently small such that We let, for this δ, φ(q) = 1 − ρ(ι(q)).Since 1 − ρ vanishes in a neighbourhood of b, there exists a geodesic ball D whose closure is disjoint from supp(1 − ρ) and therefore its double cover D is a positive distance apart from supp φ.Remark 5.4.The geometric counterpart of Lemma 5.1 is that the minimal immersion ι is unstable with respect to the area functional also if we restric to deformations that leave D (and D) fixed and that do not move M close to its singular set M \ M .We will be more specific in Section 6 below.

Relevant immersions (choice of τ )
Recall Remark 5.2.We will fix the compact subset K = M \ ι −1 (O) and will denote by K B the compact set K \ B, where B is as in Remark 5.3.Note that both K and K B are even in M , i.e. they are double covers (via ι) of compact subsets of M .We have supp φ ⊂ K B ⊂ K, for φ chosen in Lemma 5.1.Recall that φ vanishes in a neighbourhood of ∂K B (and of ∂K).We will define on K and K B suitable two-sided immersions into N , smooth up to the boundaries ∂K and ∂K B (this means that there exist open neighbourhoods of K and K B to which the immersions can be smoothly extended).
Choose c K > 0 such that c K < min (y,v)∈K σ (y,v) (by the continuity of σ > 0 on M the minimum exists and is positive).We therefore have a well-defined one-sided tubular neighbourhood of K in V M , namely K × [0, c K ), with closure contained in V M .Note that there exists an open neighbhourhood of K on which σ (y,v) > c K , by continuity of σ on M .
Recall that V M is endowed with the Riemannian metric induced by the pull-back from N .Let Π K denote the nearest point projection onto K (in coordinates, Π K (q, s) = (q, 0)).For future purposes, we ensure that c K above is also suitably small to ensure that, for x = (q, s) where |JΠ K | = (DΠ K )(DΠ K ) T and the constant C K > 0 is the maximum of the norm of the second fundamental form of ι : M → N restricted to K ⊂ M .Note that s is just the Riemannian distance of (q, s) to K (and to M ).
Choosing c0 > 0 and t0 > 0 sufficiently small, we can ensure that (q, c + t φ(q 2 ) for all t ∈ [0, t0 ] and for all c ∈ [0, c0 ].For any such c, t we thus have a smooth two-sided immersion q = (y, v) ∈ Int(K) → exp y (c + t φ(q))v from the interior of K into N .Remark 6.1.Note that, since φ = 0 in a neighbourhood of ∂K, the immersion q = (y, v) ∈ Int(K) → exp y (c + t φ(q))v agrees with q = (y, v) ∈ Int(K) → exp y (cv) in a neighbourhood of ∂K, therefore it extends smoothly to ∂K.Similarly, q = (y, v) ∈ Int(K B ) → exp y (c + t φ(q))v extend smoothly to ∂K B because φ = 0 vanishes in a neighbourhood of ∂K.Remark 6.2.(a) Again thanks to the fact that φ = 0 in a neighbourhood of ∂K, we have the following technically useful fact.For the two-sided immersion q = (y, v) ∈ K → exp y (c + t φ(q))v , with c > 0, denote by ν a choice of unit normal (which extends continuously up to ∂K) and by K c,t, φ its image.We can find c > 0 such that, for any t ∈ [0, t0 ] and c ∈ [0, c0 ], the set {exp x (sν) : s ∈ (−c, c), x ∈ K c,t, φ} is contained in K × [0, c K ).By making c smaller if necessary, we can also ensure that the set {exp x (sν) : s ∈ (− min{c, c}, min{c, c}), x ∈ K c,t, φ} is a tubular neighbourhood of K c,t, φ, in the sense that it admits a well-defined nearest point projection Π c,t onto K c,t, φ.This projection extends smoothly up to the boundary portion {exp x (sν) : s ∈ (− min{c, c}, min{c, c}), x ∈ ∂K c,t, φ}.In fact, close to {exp x (sν) : s ∈ (− min{c, c}, min{c, c}), x ∈ ∂K c,t, φ} we have that Π c,t agrees with the nearest point projection onto Γ c .
These properties essentially say that we can work with tubular neighbourhoods of K c,t, φ without interfering with the complement of F (K × [0, c K )) and it will be useful when writing Allen-Cahn approximations of the immersions q = (y, v) ∈ Int(K) → exp y (c + t φ(q))v . 6b) For notational convenience we redefine c0 , by choosing the minimum of c0 specified above and c specified in (a).Then we have a well-defined nearest point projection Π c,t : for all c ∈ (0, c0 ] and all t ∈ [0, t0 ].There exists a constant C K,c 0 ,t 0 > 0, depending only on c 0 , t 0 , on the Riemannian metric and on the C 3 norms of φ on K and of F , such that where |JΠ c,t | = (DΠ c,t )(DΠ c,t ) T and s is the distance of x to K c,t, φ.
Remark 6.3.Choosing a suitably small t 0 ≤ t0 , t 0 > 0, we can further ensure that the area of the immersion q = (y, v) ∈ Int(K) → exp y (t φ(q))v is strictly decreasing in t on the interval [0, t 0 ].This follows upon noticing that the first variation (with respect to area) at t = 0 is 0 (by minimality of M ) and the second variation at t = 0 is negative by Lemma 5.1 (see Remark 5.4).Remark that the immersions q = (y, v) ∈ Int(K B ) → exp y (t φ(q))v (the previous family of immersions restricted to Int(K B )) have the same area-decreasing property, since φ = 0 on D. For the latter family of immersions, the area at Lemma 6.1.Let t 0 be as in Remark 6.3.There exist c 0 ∈ (0, c0 ] and τ > 0 such that (i) for all c ∈ [0, c 0 ] and for all t ∈ [0, t 0 ] the area of the immersion (ii) for all c ∈ [0, c 0 ] the area of the immersion Proof.Let us prove that (i) holds for some c 0 ∈ (0, c0 ] (in place of c 0 ).Argue by contradiction: if not, then there exists c i → 0 and t i ∈ [0, t 0 ] such that the area of q ∈ ) for all i.Upon extracting a subsequence we may assume t i → t ∈ [0, t 0 ] and by continuity of the area we get that the area of q . This is however in contradiction with Remark 6.3, which says that this area is ≤ H n (K) − 2H n (B).
Let us prove that (ii) holds for some c 0 ∈ (0, c0 ] (in place of c 0 ) and for some τ > 0.
By Remark 6.3 the area of q = (y, v) ∈ Int(K) → exp y (t 0 φ(q))v is strictly smaller than H n (K).Denote by 2τ the difference of the two areas.By continuity, there exists c 0 > 0 such that for all c ∈ [0, c 0 ] the area of the immersion q = (y, v) ∈ Int(K) → exp y (c + t 0 φ(q))v is smaller than H n (K) − τ .
We will write, in Section 7, Allen-Cahn approximation of the immersions in Lemma 6.1.To that end, we will work in the tubular neighbourhoods specified in Remark 6.2.
Signed distance dist K c,t, φ .To write Allen-Cahn approximation of the immersions in Lemma 6.1 we will need to use the following notion of signed distance to K c,t, φ.Recall that φ ≥ 0 is smooth and φ = 0 in a neighbourhood of ∂K.In the coordinates of V M , K c,t, φ is identified with a graph, namely (for c ∈ [0, c 0 ] and t ∈ [0, t 0 ]) We define, on K × (0, c K ), the following "signed distance to F −1 K c,t, φ ", for c > 0. First we decide the sign of the distance: we say that (q, s) ∈ K × (0, c K ) has negative distance to F −1 K c,t, φ if s < c + t φ(q) and positive distance to F −1 K c,t, φ if s > c + t φ(q).Next we define its modulus.The modulus of the signed distance is the unsigned distance of (q, s) to is endowed with the Riemannian metric pulled back from N ).Note that if (q, s) ∈ F −1 K c,t, φ then the distance extends smoothly at (q, s) with value 0. Also remark that we do not define the unsigned distance on K × {0}.The signed distance just defined descends to a smooth function on F (K × (0, c K )) ⊂ N that we will denote by dist K c,t, φ .The set F (K × (0, c K )) is an open tubular neighbourhood of ι(K) of semi-width c K , with M removed.
7 Allen-Cahn approximations and paths in W 1,2 (N ) The overall aim in the sections that follow is to produce, for all sufficiently small ε, a continuous path in W 1,2 (N ) that starts at the constant −1, ends at the constant +1 and such that E ε is bounded by ≈ 2H n (M ) − min H n (B) 2 , τ 2 , where B and τ were chosen respectively in Remark 5.3 and Lemma 6.1 and depend only on geometric data (not on ε).Theorem 1.3 (and Theorems 1.1, 1.2) will follow immediately once this is achieved.

Choice of ε
Let B be as in Remark 5.3 and c 0 , t 0 , τ be as in Lemma 6.1.The geometric quantities H n (B) and τ are relevant in the forthcoming construction.
In the following sections we are going to exhibit, for every sufficiently small ε, a continuous path in W 1,2 (N ) with E ε suitably bounded along the whole path.We will specify now an initial choice ε < ε 1 that permits the construction of the W 1,2 -functions describing the path.When we will estimate E ε along the path, we will do so in terms of geometric quantities (typically, areas of cetain hypersurfaces, hence independent of ε) plus errors that will depend on ε.For sufficiently small ε, i.e. ε < ε 2 for a choice of ε 2 ≤ ε 1 to be specified, these errors will be ≤ C(ε | log ε |), for some C > 0 independent of ε; we will not keep track of the constants and will instead write O(ε | log ε |).At the very end (Section 8), in order to make these errors much smaller than τ and H n (B), and thus have an effective estimate on E ε along the path, we may need to revisit the smallness choice: for some ε 3 , (possibly ε 3 ≤ ε 2 ) we will get that for ε < ε 3 the errors can be absorbed in the geometric quantities.Therefore for ε < ε 3 , we will have an upper bound for E ε along the path that is independent of ε.Now we choose ε 1 .The choices of ε 2 , ε 3 will be made as we proceed into the forthcoming arguments.We restrict to ε 1 < 1, so that the O(ε 2 ) controls that we have on the approximated one-dimensional solutions in Section 2.2 are valid for all ε < ε 1 .We then require ε 1 < 1 e so to have ε | log ε | is decreasing as ε decreases so that the conditions specified on ε 1 hold also for each ε < ε 1 and, moreover, (and implicitly < 1 2 ω).Since the quantity 6ε| log ε| will appear frequently (due to the choice of truncation in Section 2.2), we will use the shorthand notation Λ = 3| log ε |, when working at fixed ε.

Allen-Cahn approximation of 2(|M | − |B|)
Recall the function G ε 0 : N → R defined in (7), which is an Allen-Cahn approximation of ι : M → N , i.e. a W 1,2 function with nodal set close to the image of ι and such that its Allen-Cahn energy E ε (G ε 0 ) is approximately 7 the area of ι (i.e.≈ 2H n (M )).Due to the fact that we replace hypersurfaces by non-sharp transitions, the function G ε 0 can also be thought of as an Allen-Cahn approximation of Γ 2 ε Λ (that is exactly the nodal set of G ε 0 ).Definition of f .We will now "remove the ball B" from G ε 0 : N → R. In other words, we will write an Allen-Cahn approximation f of 2(|M | − |B|), or, equivalently, 7 In Section 4 we only established an upper bound for Eε(G ε 0 ), and most of the times an upper bound is all that will matter for our Allen-Cahn approximations (although a lower bound in terms of the area of the correspoding immersion is also going to be always valid).In the case of G ε 0 , such a lower bound for Eε(G ε 0 ) will be established later.
of ι| M \ B .Always because we have non-sharp transitions, we can think of f also as an Allen-Cahn approximation of Γ 2 ε Λ with two balls removed.Although f = f ε does depend on ε, we drop the ε for notational convenience.What is important to keep in mind is that we can perform the contruction of f given below for any ε < ε 1 and that we will obtain estimates on E ε (f ε ) that are uniform in ε.
To this end, we let χ ∈ C ∞ c ( M ) be smooth and even (i.e.χ(p) = χ(q) if ι(p) = ι(q)) such that χ = 1 on B, |∇χ| ≤ 2 R , where R is the radius of B, and suppχ ⊂⊂ D. Then we define, using coordinates (q, s) where Ψ t is as in (3).Since χ is even, the function G ε 0,B descends to a well-defined function f on F (K × [0, c K )) (this is a tubular neighbourhood of semi-width c K around ι(K)).Note that f agrees with G ε 0 on F (K \ D) × [0, c K ) and on F ((K × (c K /2, c K )) (on the latter both are equal to −1), therefore we extend f to N by setting it equal to and obtain that , we will in fact conclude that f is W 1,∞ on N .We only need to check it around points x ∈ D. Let χ 0 : M → R be defined by χ 0 (y) = χ(ι −1 (y)); this is a smooth function compactly supported in D. In a neighbourhood of x ∈ D we can choose a small geodesic ball B r (x) ⊂ M and use Fermi coordinates (y, a) ∈ B r (x) × (−c K , c K ).Then in this neighbourhood f (y, a) = Ψ 4 ε Λχ 0 (y) (a), hence f is Lipschitz on B r (x) × (−c K , c K ).(The Jacobian factor that measures the distortion of the Riemannian metric from the product metric on B r (x) × (−c K , c K ) is bounded by a constant that only depends on the geometric data F (K) ⊂ M ⊂ N ; therefore it suffices to observe that Ψ 4 ε Λχ 0 (y) (a) is Lipschitz with respect to the product metric.)Therefore f ∈ W 1,∞ (N ).
Allen-Cahn energy of f .To estimate from above the Allen-Cahn energy of f , since 9), we only need to compute the energy of f on F (D × [0, c K ]) (and, similarly, the energy of G ε 0 on F (D × [0, c K ])).We can therefore use coordinates (q, s) on D × [0, c K ] ⊂ V M as in (14) and apply the coarea formula (for the function Π K (q, s) = (q, 0), whose Jacobian determinant |JΠ K | is computed with respect to the Riemannian metric induced from N ): The notation ∇ q stands for the gradient projected onto the level sets of s.By definition of G ε 0,B we have, in D × (0, c K ): 4 ε Λ∇ q χ and since (R denotes the radius of (Here C = (8 • 6) 2 .)Since B, D, R and C are independent of ε, (17) implies that the third term on the right-hand-side of ( 16) can be made arbitrarily small by choosing ε sufficiently small; this term is The first term on the right-hand-side of ( 16) vanishes because G ε 0,B = −1 on that domain.For the second term on the right-hand-side of ( 16), note that the inner integral only gives a contribution in [0, Recalling the bounds on the Jacobian factor |JΠ K | given in ( 12) and the energy estimates on the one-dimensional profiles, see ( 1) and ( 3), we find second term on right-hand side of ( 16) We can thus rewrite ( 16) as a leading term 2σ H n ( D) − H n ( B) plus errors; for a sufficiently small choice of ε 2 ≤ ε 1 , for ε < ε 2 , all errors are of the type O(ε | log ε |).
We therefore conclude that the following estimate holds for all ε < ε 2 : Going back to G ε 0 , we can give a lower bound to its energy on F (D × [0, c K ]) with a computation analogous to the one just carried out.With coordinates (q, s) ∈ D×[0, c K ] we have that G ε 0 is simply the function Ψ(s) and therefore |∇G ε 0 | is given by ∂ ∂s Ψ(s) (the gradient is parallel to the ∂ ∂s ).Using the coarea formula (again8 with Π K ) we get that where we used ( 12) and ( 1) and ( 3).The result in ( 18) is of the form 2σH n ( D) plus errors.The errors are of the form O(ε | log ε |) for all ε < ε 2 for some suitably small choice of ε 2 ≤ ε 1 .
Remark 7.1 (on the choice of ε 2 ).We make the choice of ε 2 several times along the construction, always with the scope of making the errors controlled by C ε | log ε | with C independent of ε ∈ (0, ε 2 ).The specific value ε 2 might change from one instance to the next, but since we make finitely many choices we implicitly assume that the correct ε 2 is the smallest of all.From now on, this remark will apply every time we say that the errors are of the form O(ε | log ε |) for all ε < ε 2 for some suitably small choice of ε 2 .
In conclusion for all ε < ε 2 we have that Recall that f does depend on ε, although we are not expliciting the dependence for notational convenience, and that we can produce f (as defined above) for every ε < ε 1 .By ( 9) and ( 19), and the fact that ), we conclude that for a sufficiently small choice of ε 2 ≤ ε 1 , for all ε < ε 2 , the following estimate holds: This says that f is a good 9 Allen-Cahn approximation of 2(|M | − |B|).In terms of the immersions of Lemma 6.1, f is also an Allen-Cahn approximation of q = (y, v) ∈ Int(K B ) → exp y (2 ε Λv) (the nodal set of f contains the image of this immersion with boundary).

From
In this section we construct a continuous path in W 1,2 (N ) that joins f to the constant −1, keeping E ε along the path controlled by E ε (f ).
We begin by introducing the following one-parameter family of functions: for r ∈ where Ψ r is as in (3).Since H We compute E ε (Y ε 0 ) by using the coarea formula (slicing by the distance function d M , for which |∇d M | = 1) as we did for G ε 0 (see ( 9)).We obtain using ( 1) and the fact that r ) → 0 as r → 4 ε Λ.Now we give a lower bound for the energy of Y ε r on the domain F (D × [0, c K )) as we did for G ε 0 in (18), i.e. using the coarea formula for the function Π K .Note that on this domain we can use the coordinates (q, s) on D × [0, c K ) and the fact that the gradient of Y ε r is parallel to ∂ ∂s .We have where we used (12) and the fact that the part in paretheses of the inner integrand is independent of q.We therefore conclude, from ( 22) and ( 23), the following estimate for the Allen-Cahn energy of Y ε r in the complement of F (D × [0, c K )): there exists ε 2 ≤ ε 1 sufficiently small such that for all ε < ε 2 Definition of the path f r .We now define a continuous path r ∈ [0, 4 ε Λ] → f r ∈ W 1,2 (N ) as follows.Recalling the definition of χ ∈ C ∞ c ( M ) and using coordinates (q, s) where Ψ t is as in (3).The function f r : N → R is then defined by Note that f r is well-defined on D since χ is even.Remark also that for r = 0 this function is f and for r = 4 ε Λ it is the constant −1.Moreover, f r ∈ W 1,∞ (N ) for every r.To see this, notice that Y r,B is smooth on D × (0, c K ), so f r is smooth on We thus only need to check that f r is Lipschitz locally around any point x ∈ D. Using Fermi coordinates (y, a) ∈ B(x)×(−δ, δ), where B(x) is a small geodesic ball in M centred at x and δ > 0, we have the following expression for f , thanks to the fact that Ψ r : R → R is even for every r: f r (y, a) = Ψ 4 ε Λχ 0 (y)+r (a), and since χ 0 is smooth, we obtain that f r ∈ W 1,∞ on the chosen neighbourhood of x. (As we did in (15), we use the fact that being Lipschitz for the product metric on B(x) × (−δ, δ) implies Lipschitzianity for the Riemannian metric induced from N .)In conclusion we have N ) is moreover continuous.Let us check the continuity of ∇f r in r (with respect to the L 2 topology on N ).For ∇f r on F ( D×[0, c K )) we have the following expression, using (q, s)-coordinates on D × (0, c K ): 4 ε Λ∇χ(q)Ψ 0 (s + 4 ε Λχ(q) + r), Ψ 0 (4 ε Λχ(q) + r + s) .
By continuiuty of translations in L p and smoothness of χ we get that ∇f r is continuous in r (with respect to the L 2 -topology, or even L p for any p).Similarly we can argue for , where f = Y ε r,0 and the gradient is Ψ 0 (r + d M (x))∇d M (x): this changes continuously with r (with respect to the L 2 -topology, or even L p for any p).Therefore we have that r ∈ [0, 4 ε Λ] → ∇f r ∈ L 2 (N ) is continuous.The fact that f r changes continuously in r with respect to the L 2 topology is even more straightforward.
Energy along the path.To estimate E ε (f r ) we compute the energy on F ( D ×[0, c K )) using the coarea formula for Π K , similarly to (20), in the coordinates (q, s) ∈ D×[0, c K ).Notice that Y ε 0,B (q, s) = −1 for q ∈ B. Then we obtain ≤ (1 + 8 ε Λ) and by the estimate in (24) we conclude that there exists ε 2 ≤ ε 1 such that for all ε ≤ ε 2 the following estimates hold for r ∈ [0, 4 ε Λ]: (The second follows from the first since the energy of The second estimate shows the uniform energy control on r ∈ [0, 4 ε Λ]; the first shows that E ε (f r ) → 0 as r → 4 ε Λ. Remark 7.2.At least for n ≤ 6 it is possible to produce a continuous path from f to −1, with a similar energy control as in (26), by employing an alternative argument.One can employ the negative E ε -gradient flow starting at a suitably constructed function f 2 that is W 1,2 -close to f and with E ε (f 2 ) ≈ E ε (f ).The flow will be mean convex and will tend to the constant −1, reaching it in time O(ε | log ε |).The ε → 0 limit of such paths is then the Brakke flow that starts at 2(|M | − |B|) and vanishes instantaneously.The family (25) that we gave in this section mimics exactly this flow, however it is more elementary, even for n ≤ 6, as we can exhibit the path explicitly (and moreover presents no additional difficulties for n ≥ 7).Note that the path f r that we produced also reaches −1 in time O(ε | log ε |).

Lowering the peak
In this section we construct the next portion of our path, starting at f .The immersions in Lemma 6.1 are particularly relevant, as they provide the geometric counterpart of this portion of the W 1,2 -path: first we use the immersions in (i) of Lemma 6.1 keeping c = 2 ε Λ and increasing t from 0 to t 0 ; then we connect the final immersion just obtained to the one in (ii) of Lemma 6.1 with t = t 0 and c = 2 ε Λ (in doing so, we "close the hole at B").The portion of the path that we exhibit in this section is made of Allen-Cahn approximations of the immersions just described.It is this portion of the path that "lowers the peak" of E ε , keeping it a fixed amount below 2H n (M ) (thanks to the estimates in Lemma 6.1).
We will keep using the shorthand notation Λ = 3| log ε |.All the functions that we will construct in this section coincide with G ε 0 in the complement of F (K × [0, c K )).By construction they will in fact agree with G ε 0 in a neighbourhood of ∂F (K × [0, c K )) (guaranteeing a smooth patching) and thanks to Remark 6.2 and since 2 ε Λ < c 0 /20 (Section 7.1) we can use tubular neighbourhoods of semi-width 2 ε Λ around K c,t, φ for every c ≥ 2 ε Λ to define Allen-Cahn approximations of the immersions in Lemma 6.1.
Recall the notation K c,t, φ from Section 6: it denotes the image via F : V M → N of the graph {(q, s) ∈ V M : q ∈ K, s = c + t φ(q)}, for t ∈ [0, t 0 ] and c ∈ [0, c 0 ].In other words, K c,t, φ is image of the immersion (smoothly extended up to ∂K, see Remark 6.1) q = (y, v) ∈ K → exp y (c + t φ(q))v .Recall the definition of signed distance provided in Section 6 and denote by dist K c,t, φ the signed distance to K c,t, φ, well-defined Then the definition of f in ( 14)-( 15), can equivalently be given as follows where and, with a slight abuse of notation, Π K (x) is the nearest point projection of x onto M .(In the coordinates of V M we have Π K (q, s) = (q, 0), which is the notation used in Section 6; the map on F (K × [0, c K )) that we are using above should then be F • Π K • F −1 , we however denote both the map in K × [0, c K ) and the map in We point out the following facts.Let x ∈ F (K × {0}) and x j → x, x j ∈ K × (0, c K ) (so that the signed distance is negative on x j for j sufficiently large).Then In particular, this continuous extension is valid on (an open neighbourhood of) D.
Definition of g t .We construct now the portion of path t ∈ [0, t 0 ] → g t ∈ W 1,2 (N ) whose geometric counterpart is given by the immersions in (i) of Lemma 6.1 with c = 2 ε Λ and t ∈ [0, t 0 ].These immersions "have a hole at B".We set, for t ∈ [0, t 0 ]: In the second line of (27) we are using the fact that dist K 2 ε Λ,t, φ is well-defined and continuous on D, with value −2 ε Λ (see Remark 7.3).Also note that on the definition in the second line agrees with the definition of G ε 0 ( φ vanishes in a neighbourhood of ∂K, see Section 6) and the same is true on F (K × {c K }) (g t = −1 in a neighbourhood).For t = 0 we have g 0 = f , by the expresion of f given earlier in this section.
g t ∈ W 1,∞ (N ) for each t.Let us check first that g t it is continuous on N for each t.In view of the comments just made, this needs to be checked only at an arbitrary x in F ((Int(K) \ D) × {0}).Let x j → x, then for sufficiently large j we have To check that g t ∈ W 1,∞ (N ), note first that the definition in the second line of ( 27) is equal to the one of G ε 0 in a neighbourhood of the boundary of )), and actually even in a neighbourhood of the boundary of F (K × [0, c K )).Moreover, for x ∈ B we have Therefore we only need to check that g t is locally Lipschitz around points x ∈ F ((Int(K) \ B) × {0}).We distinguish two cases.If x / ∈ D, i.e. if x ∈ F ((Int(K) \ D) × {0}), then g t is actually C 1 in a neighbourhood of x.This is seen by repeating the argument used above (for the continuity of g t at such point) to prove that |∇g t (x j )| → 0 (using the fact that H ε is smooth on R and equal to 0 on [2 ε Λ, ∞)).We therefore have: ) with constant value 1 and ∇g t extends continuously to this set with value 0. From these facts it follows that the L ∞ function equal to ∇g t on F (( is the ditributional derivative of g t in a neighbourhood of x and therefore g t is C 1 in a neighbourhood of x.In the second case, i.e. if x ∈ D \ B, then for a sufficiently small ball and D are disjoint) and we can use a well-defined system of Fermi coordinates (y, a) ∈ B ρ (x) × (−c K , c K ).In these coordinates we have dist , which is Lipschitz in the neighbourhood.The path t → g t is continuous.It suffices to check that the second line in (27) is continuous in t.The proof is analogous to the one where we proved that f r defined in (25) is continuous in r; it can be carried out using the coordinates on V M and the fact that the graph {(q, s) : q ∈ K, s = 2 ε Λ + t φ(q)} changes smoothly in t, hence so does the function dist Energy of g t .To give an upper bound for E ε (g t ) we first need a lower bound for the energy of G ε 0 on F (K × [0, c K )).This is analogous to the estimate in (18): where we used (1).From ( 9) and ( 28) we obtain that, for some suitably small choice of ε 2 ≤ ε 1 , for all ε < ε 2 the following holds for the energy of G ε 0 (and thus also of We now pass to an estimate for the energy of g t in F (K × [0, c K ]).For this we will use Fermi coordinates for a tubular neighbourhood of K 2 ε Λ,t, φ of semi-width 2 ε Λ. Denote by (y, a) ∈ K 2 ε Λ,t, φ × (−2 ε Λ, 2 ε Λ) such coordinates and by Π 2 ε Λ,t the nearest point projection from the chosen tubular neighbourhood onto K 2 ε Λ,t, φ (see Remark 6.2).Remark that g t = −1 on F ( B × [0, c K )) so there is no energy contribution in this open set.The coarea formula (for the function Π 2 ε Λ,t ) then gives: Therefore for some suitably small choice of ε 2 ≤ ε 1 , for all ε < ε 2 the following holds is the image of K B via the immersion in (i) of Lemma 6.1 when c = 2 ε Λ.Using Lemma 6.1 in the last estimate and putting it together with (29) we finally obtain that, for some suitably small choice of ε 2 ≤ ε 1 , for all ε < ε 2 the following estimate holds for all t ∈ [0, t 0 ]: Definition of g t 0 +r : "closing the hole at B".We have constructed a continuous path t ∈ [0, t 0 ] → g t ∈ W 1,2 (N ) with g 0 = f and with E ε uniformly controlled by (30).The next portion of the path will start from g t 0 and will "close the hole at B". On the geometric side, we are starting at the immersion in Lemma 6.1 (i) with c = 2 ε Λ and t = t 0 , and ending at the immersion in Lemma 6.1 (ii) with c = 2 ε Λ and t = t 0 .We define for r ∈ [0, 1] .
The fact that g t 0 +r ∈ W 1,∞ (N ) for every r ∈ [0, 1] follows by repeating the arguments used for g t , where the only part that has to be altered is the local expression of g t 0 +r around points of D. Using Fermi coordinates (y, a) with y ∈ D, a ∈ (−c K , c K ) we get g t 0 +r (F (y, a)) = H ε 4 ε Λ(1−r)χ(y) (−|a| + 2 ε Λ), which is Lipschitz.Notice that this is the domain in N where we are "closing the hole": when r = 1 the expression just obtained becomes g t 0 +1 (F (y, a)) = H ε 0 (−|a| + 2 ε Λ) = Ψ 0 (a) and so Note also that r ∈ [0, 1] → g t 0 +r ∈ W 1,2 (N ) is a continuous path (with a proof as the ones used earlier for g t and f r ).
Energy of g t 0 +r .We use the coarea formula as we did to reach (30).We get Therefore for some suitably small choice of ε 2 ≤ ε 1 , for all ε < ε 2 the following holds Note that K 2 ε Λ,t 0 , φ is the image of K via the immersion in (ii) of Lemma 6.1 when c = 2 ε Λ.Using Lemma 6.1 in the last estimate and putting it together with (29) we finally obtain that, for some suitably small choice of ε 2 ≤ ε 1 , for all ε < ε 2 the following estimate holds for all r ∈ [0, 1]:
Lemma 7.1.For all sufficiently small ε there exists a smooth function m : Proof.From (8) we have for all sufficiently small ε that where µ ε > 0 is defined in (34).Recall that this inequality means that (the positive Radon measure) Therefore we can find a sufficiently small ρ 0 > 0 (depending on ε, in fact we may choose ρ 0 ≈ ε 2 ) such that for all sufficiently small ε we have Let C N be the constant in Lemma A.2.We are going to work with ε sufficiently small to ensure (in addition to the previous conditions identified so far in this proof) that 2 ε C N < µ ε /20.From now we work at fixed ε (satisfying the smallness conditions just imposed).Let η δ be the mollifiers defined in Appendix A for δ < δ 0 , where δ 0 > 0 depends only on the geometry of N .Then the (smooth) function −F ε,µ ε (G ε 0 −ρ 0 ) η δ defined in (57) is positive for all δ, more precisely for all sufficiently small δ (one needs µε 12 > O(δ 2 ), where O(δ 2 ) appears in (52)) This follows from (35) and ( 52), (57).We now mollify (G ε 0 − ρ 0 ) as in (53).We have From Lemma A.1 (a) we obtain that the functions (G ε 0 − ρ 0 ) η δ converge uniformly on N to (G ε 0 − ρ 0 ) as δ → 0. Therefore (for δ sufficiently small −2 < (G ε 0 − ρ 0 ) η δ < 2 since the same bound holds for By the triangle inequality we therefore have as δ → 0. By Lemma A.2 there exists C N (depending only on the geometry of N ) such that for all δ < δ 0 we have ∆(( Therefore the modulus of the difference of the following two (smooth) functions , where the infinitesimal of δ is given by the norm in (37) plus O(δ 2 )µ ε .Recall (36) and the smallness condition imposed on ε.Then for sufficiently small δ, writing m = (G ε 0 − ρ 0 ) η δ , we have Finally note that for sufficiently small δ we also have m < g t 0 +1 , since G ε 0 − ρ 0 < g t 0 +1 and (G ε 0 − ρ 0 ) η δ converges uniformly to G ε 0 − ρ 0 as δ → 0 (Lemma A.1).
Remark 7.4.(choice of ε 2 , again) We will assume that Lemma 7.1 is valid for all ε < ε 2 , where once again we change the choice of ε 2 if necessary.Flow from m.We consider now the negative F ε,µ ε -gradient flow with initial condition given by the smooth function m, i.e. the solution m t to the PDE where ∆ is the Laplace-Beltrami operator on N .This semilinear parabolic problem has a solution for t ∈ [0, ∞) and m t ∈ C ∞ (N ) for all t > 0, as we will now sketch.Short-time existence and uniqueness for a weak solution in W 1,2 (N ) are valid by standard semilinear parabolic theory (rewrite the problem as an integral equation, then use a fixed point theorem).To see why we get global existence in our case, integrate (39) on any interval [0, T ] on which the weak solution is defined: we get With our choice of W that is quadratic on (±2, ±∞) we can ensure that W (u) ε − µ ε u is bounded below.Then (40) gives a priori bounds N |∇m t | 2 ≤ C m 0 ,ε,W independently of t ∈ [0, T ].Again from (40), moving the term 1 ε N (W (m t ) − ε µ ε m t ) to the left-handside and recalling |u| 2 ≤ C W,ε max{2, W (u)  ε − µ ε u}, we also get an a priori L 2 -bound on m t .In conclusion with C independent of t.This first bound in (41) provides the assumption under which short-time existence can be iterated to lead global existence for a weak solution to (39) in W 1,2 (N ).

Lemma 7.2 (mean convexity of m t ). The positivity condition
Proof.For notational convenience, we write for this paragraph F t = ε ∆m t − W (mt) ε +µ ε (right-hand-side of the first line in (39)).By the previous discussion, F t is smooth on N for all t ∈ [0, ∞).Differentiating F t = ε ∆m t − W (mt) ε + µ ε (and using ε ∂ t m t = F t ) we get the evolution of F t , given by ∂ γ, and the constant γ = 0 is also a solution to the same PDE.The condition F t > 0 is therefore preserved by the maximum principle, since it holds at t = 0 by Lemma 7.1.Lemma 7.2 implies in particular that m t : for all t by the maximum principle).Therefore W (m t ) → W (m ∞ ) in W 1,2 -weak.By the second bound in (41) we have L 1 -summability in time, on t ∈ (0, ∞), for ∂ ∂t m t L 2 (N ) and therefore there exists t j → ∞ such that the function ∂ ∂t m t : N → R has L 2 (N )-norm that tends to 0 along the sequence t j .These facts imply that the weak formulation of the PDE in (39) passes to the limit as t j → ∞ and gives that m ∞ solves −F ε,µ ε = 0 in the weak sense.Standard elliptic theory (or passing parabolic estimates for m t to the t → ∞ limit) then show that m ∞ ∈ C ∞ solves −F ε,µ ε = 0 strongly.Lemma 7.3 (stability of m ∞ ).The limit m ∞ of the flow m t (as t → ∞) is a stable solution of F ε,µ ε = 0.
Proof.This is a consequence of the "mean convexity" of m t (Lemma 7.2) and of the maximum principle.We give the explicit argument.Recall from the previous discussion that m ∞ is stationary, i.e.F ε,µ ε (m ∞ ) = 0. Also recall that the second variation at u : N → R of the functional F ε,µ (for a constant µ) on the test function φ is given by the quadratic form Q(φ, φ) = N ε |∇φ| 2 + W (u) ε φ 2 (the term involving µ disappears because it is linear) and the Jacobi operator is given by − ε ∆φ + W (u) ε φ.Let ρ 1 be its first eigenfunction (with respect to the energy F ε,µ ε ), then ρ 1 is (strictly) positive and smooth on N .Consider, for s ∈ (−δ, δ) (for some small positive δ), the functions m ∞ − sρ 1 .Then their first variation satisfies If m ∞ were unstable, then the first eigenfunction would satisfy − ε ∆ρ 1 + W (m∞) ε ρ 1 = λ 1 ρ 1 for some λ 1 < 0 and therefore on N .Then we could choose s 0 > 0 sufficiently small so that we have m t > m 0 , in particular m ∞ > m 0 .Choose s sufficiently small so that s < s 0 and m ∞ − sρ 1 > m 0 .Let τ > 0 be the first time for which m τ has a point x such that m τ Proposition 7.1.If Ric N > 0 then any stable solution to F ε,µ = 0 on N must be a constant (here µ can be any constant.) Proof.Let u be a stable solution to F ε,µ (u) = 0. We test the stability inequality Q(•, •) ≥ 0 on a test function of the form |∇u|φ for φ ∈ C 2 (N ).We get (this follows from the expression of Q given above by using Bochner's identity, see [5], [24]) where A ε stands for the Allen-Cahn second fundamental form of u and We plug in φ = 1 so the positiveness of Ric N gives ∇u ≡ 0.
Lemma 7.3 and Proposition 7.1 give that m ∞ is a constant.There exist exactly two constant solutions of F ε,µ = 0: for this, the constant k must satisfy W (k) = ε µk (and therefore W (k) ≈ c 2 w ε 2 µ 2 ), so one constant is slighly larger than −1 and the other is slighly larger than +1 when ε is sufficiently small (both solutions are trivially checked to be stable).In our case, since m ∞ > m 0 and m 0 > 1/2 on an open neighbourhood of M , we conclude that m ∞ is the constant slightly larger than +1, which we will denote by k µε : Flow from g t 0 +1 .We are now ready to consider the negative F ε,µ ε -gradient flow {h t } starting at h 0 = g t 0 +1 .We first make the initial datum smooth, by considering mollifiers η δ for δ ∈ (0, δ] as in Appendix A and δ sufficiently small to preserve the strict inequality with m, i.e. to ensure g t 0 +1 η δ > m for δ ∈ (0, δ].The family is continuous in δ and extends by continuity at δ = 0 with value g t 0 +1 (see Remark A.1). Continuity is also valid for δ ∈ (0, δ] → g t 0 +1 η δ ∈ C 0 (N ).As a consequence, E ε (g t 0 +1 η δ ) varies continuously with δ and therefore, upon choosing δ possibly smaller, we also have, in addition to (44) and to g t 0 +1 η δ > m, that the following holds for all δ ∈ (0, δ], We now let h 0 = g t 0 +1 η δ be the initial condition for the negative F ε,µ ε -gradient flow: By the maximum principle, since m 0 < h 0 , the two flows (39) and (46) preserve m t < h t for all t.11Since g t 0 +1 ≤ 1 by construction, we also have h 0 < k µε , therefore h t < k µε for all t > 0 by the maximum principle.On the other hand we saw that m t → k µε as t → ∞, therefore (with smooth convergence, in particular we have continuity in t for Evaluation of E ε on the path h t .Let us estimate the value of E ε along this path.For this, note that F ε,µ ε is decreasing along the flow {h t }, therefore E ε (h t ) ≤ E ε (h) + 2µ εH n+1 (N ) for all t (where we used h t < 2 for all t).This implies that E ε is bounded above indepedently of ε; more precisely, recalling that In other words we obtain, for ε 2 ≤ ε 1 sufficiently small, the upper bound for all t and for all ε < ε 2 .
To complete the path, we will now connect h ∞ = k µε to +1 by a negative E ε -gradient flow: Since E ε (k µε ) = −µ ε < 0 it follows that mean-convexity is preserved and k t decreases in t.Moreover +1 is a solution and thus acts as a lower barrier.In conclusion the negative E ε -gradient flow starting at k µε tends to +1. (One can check that in fact k t is a constant for each t.)The energy E ε (k t ) is decreasing, so the same upper bound that we had in (48) holds: for all t and for all ε < ε 2 .
8 Conclusion of the proof of Theorems 1.3, 1.1 and 1.2 In the previous sections we exhibited (given M as in Theorem 1.3, which also fixed B and τ by Remark 5.3 and Lemma 6.1) for all sufficiently small ε (namely ε < ε 2 ) the following five continuous paths in W 1,2 (N ): (25) reversed, ( 27), (31), (44), (46).In the order just given, these paths have matching endpoints, therefore their composition in the same order provides a continuous path in W 1,2 (N ) for all ε < ε 2 , that starts at the constant −1 and ends at the constant +1 and such that thanks to ( 26), (33), ( 45), (48), (50).Choosing ε 3 sufficiently small to ensure that the above bound gives, for all ε < ε 3 , that the maximum of E ε on the path is at most 2H 2 .The path is in the admissible class for the minmax construction in [10], therefore the maximum on this specific path controls from above the minmax value c ε achieved by the index-1 solution u ε obtained from [10] (for all ε < ε 3 ).Summarising, for every M ⊂ N as in Theorem 1.3 there exist ε 3 > 0, τ > 0 and B ⊂ M (non-empty) such that for all ε < ε 3 This concludes the proof of the strict inequality in Theorem 1.3.
For Theorem 1.1 it suffices to observe that the integral varifold V produced in [10] is (thanks to [27], [25]) such that each connected component of reg V (the smoothly embedded part of spt V ) has the properties needed so that it can be used in place of M in Theorem 1.3, or in (51) above; moreover, the mass V (N ) of V is lim ε i →0 c ε i (see Section 2.1).Letting M be any connected component of reg V and denoting by θ ∈ N its (constant) multiplicity, using (51) we get θH n (M ) ≤ V (N ) < 2H n (M ).This implies θ = 1 and the multiplicity assertion in Theorem 1.1 is proved.
The fact that the minimal hypersurface is two-sided then follows immediately, since under multiplicity-1 convergence (and by the lower energy bounds in [10]) we have that , where u ∞ is a non-constant function that takes values in {−1, +1} and, moreover, V is the multiplicity-1 varifold associated to the reduced boundary of the set (of finite perimeter) {u ∞ = +1} (there is no "hidden boundary" in the limit).We therefore have a global normal on reg V (the interior-or the exerior-pointing normal for ∂{u ∞ = +1}).Theorem 1.2 is therefore proved.Remark 8.1.Note that reg V has to be connected, since each connected component of it is unstable (because it is two-sided and Ric N > 0) and therefore the Morse index of reg V is at least the number of its connected components.On the other hand, by [11] the Morse index of reg V is ≤ 1.An alternative argument for the connectedness, that does not rely on two-sidedness, can be given by means of the maximum principle for stationary varifolds ( [13] [27]) and the Frankel property 12 for Ric N > 0 (using the regularity of V that follows from [25] and [27]).

A Mollifiers
We explain in detail the mollification procedure used in Section 7.5.For this appendix, notation is reset.Let (N, g) be a closed Riemannian manifold of dimension n + 1 and f : N → R in W 1,∞ (N ).We are going to produce, for every δ > 0 sufficienly small, a smooth function f δ : N → R such that f δ → f strongly in W 1,2 (N ) as δ → 0 (even W 1,p for every p < ∞, but we will not need this).The function f δ is defined as a convolution f η δ , for a suitable mollifier η δ .Moreover we will check that, if additionally ∇f ∈ BV (N ), then we have, for all δ sufficiently small, that ∆ f δ = (∆f ) η δ + E δ , where (∆f ) η δ is the convolution of the Radon measure ∆f with the mollifier η δ and hence it is identified with its (smooth) density with respect to H n+1 , and E δ is a smooth function bounded in L ∞ by a constant that only depends on N .It would not suffice for our scopes in Section 7.5 to have a convolution procedure that gives ∆ f δ → ∆f as measures, therefore we give an ad hoc contruction here.
We begin with the definitions.The standard smooth mollifier on R is η(x) = e for |x| < 1, and η(x) = 0 for |x| ≥ 1.In the following, δ < inj(N ).We then let η δ : N × N → R be defined as η δ (x, y) = 1 0 η(s)s n ds, where the integration is with respect to the Lebesgue (n + 1)-dimensional measure.Therefore for every x, using normal coordinates centred at x, the function η δ • exp x integrates to 1 in the ball of radius δ in the tangent space to N at x, endowed with the Euclidean metric.Moreover, denoting by B δ (x) the geodesic ball centred at x, there exist δ 0 < inj(N ) and C N > 0 such that, for all x ∈ N and for all δ ≤ δ 0 , we have where |O(δ 2 )| ≤ C N δ 2 .The constant C N depends ony on the curvature of N , more precisely on the maximum of the modulus of the sectional curvature (recall that N is compact).
Choice of δ 0 .The sectional curvatures of N are bounded in modulus since N is compact.Recalling Riccati's equation and the Bishop-Günther inequalities (see the final inequality in the proof of [8,Theorem 3.17], combined with [8, (3.23)] in the case P = {x}) there exist δ 0 < inj(N ) and C N > 0 such that for all x ∈ N and for δ ≤ δ 0 we have |H n (∂B δ (x)) − (n + 1)ω n+1 δ n | ≤ C N (n + 1)ω n+1 δ n+2 , where ω n+1 is the Euclidean volume of the unit ball in R n+1 .At the same time we can also ensure the following (see [8, (3.35)], or also [9,Lemma 12.1]).For all x ∈ N and for δ ≤ δ 0 , denoting by H x,δ the mean curvature function on the geodesic sphere of radius δ around the point x (with respect to the outward-pointing normal, hence H x,δ ≤ 0) we have (− n δ is the Euclidean mean curvature of the sphere of radius δ in R n+1 ) H x,δ + n δ ≤ C N δ on ∂B δ (x).
Proof of (52).This follows by using the coarea formula in B δ (x) for the function d(x, •), for which |∇d(x, •)| = 1.By the choice of δ 0 above we have C N > 0 such that for all x ∈ N and for s ≤ δ 0 , |H n (∂B s (x))−(n+1)ω n+1 s n | ≤ C N (n+1)ω n+1 s n+2 .Then using the coarea formula we get B δ (x) η δ (x, y)dH n+1 (y) = 1 This follows by writing, as done for Lemma A.1 (i), N |q(y) − q(x)|η δ (x, y)dH n+1 (y) = N |q(y)−q(x)| η δ (x,y) 1+O(δ 2 ) dH n+1 (y)+ N |q(y)−q(x)| O(δ 2 ) 1+O(δ 2 ) η δ (x, y)dH n+1 (y).The second term tends to 0 as argued earlier.The first term tends to 0 if x is a Lebesgue point of q (hence for almost all x).Then we have, with ∇f • v in place of q:  (In the last term, we have included H n+1 ( Ũ ) ≤ H n+1 (N ) in the constant CN .)The braced integrand in the first term tends to 0 for a.e.x by (55).Moreover, the braced expression is bounded for every x by 4 ∇f 2 L ∞ (N ) (1 + O(δ 2 )) 2 , which is summable on N .Hence we can use dominated convergence to conclude that the first term tends to 0 as δ → 0. The second tends to 0 a well, therefore we conclude that Step 2. We compute the difference between the two (smooth) functions (∇f • v) η δ and ∇(f η δ ) • v and prove that it goes to 0 uniformly on Ũ .We work in normal coordinates centred at an arbitrary point O ∈ Ũ , namely in the ball D = {x ∈ R n+1 : |x| < δ 0 /2}, with exponential map exp O : D → B δ 0 /2 (O) ⊂ U .We will evaluate the difference of the two functions at O, making sure that the result does not depend on O. Since we are interested in ∇(f η δ ) • v, we need to let x vary in a neighbourhood of O before evaluating the derivative, therefore we will assume x ∈ {x ∈ R n+1 : |x| < δ 0 /4} and δ < δ 0 /4, so that y stays in D.
We use the customary notation g ij for the metric coefficients, |g| for the volume density induced by g.We denote by h the Lipschitz function on D given by f • exp O : D → R and by ρ : D × D → R the mollifier ρ(x, y) = η δ (exp O (x), exp O (y)), for an arbitrary δ < δ 0 4 .We point out that ρ(0, y) = 1 cnδ n+1 η |y| δ because we are in normal coordinates, where | • | denotes the Euclidean length.We write ∇ g to denote the metric gradient in D, (∇ g ) i = g ij ∂ x j .Let v be represented, in the chart, by v j ∂ j .We fix an arbitrary and let v = (v 1 , . . ., v n+1 ) = (v 1 , . . ., v n+1 ).We will write • between two vectors to denote the scalar product induced by g, so ∇ g h • v = g ij g ia ∂ xa h v j = δ a j ∂ xa h v j = ∂ x j h v j (= dh(v)).We restrict to x ∈ D δ 0 /4 and we compute for δ < δ 0 4 the coordinate expression for ∇(f η δ ) • v (integration is in dy unless otherwise specified): Consider the first term from the last line, recalling that v j (x) multiplies I in (56): Having rewritten the first term in the last line, we evaluate (56) at x = 0 to obtain It is immediate that IV | x=0 = 0.In V we have ρ(0, y) = 0 for d(0, y) = |y| ≥ δ, therefore |v j (0) − v j (y)| ≤ C|y| for some constant C that depends on derivatives of v in U and can be thus chosen independently of U (there are finitely many U 's) and of v (finitely many smooth vector fields).We therefore get that V | x=0 is bounded in modulus by C ∇f L ∞ δ(1 + O(δ 2 )) ≤ C ∇f L ∞ δ for some C that depends only on the choices of charts and vector fields.In III| x=0 , the integrand is non-zero only for |z| ≤ δ.Let CN > 0 be an upper bound for the modulus of the second derivatives of the volume element in a normal coordinate system of radius δ 0 centred at an arbitrary point in N (such a constant exists by the compactness of N , the smoothness of the metric and the fact that δ 0 < inj(N )).Recalling that in normal coordinates the metric coefficients have vanishing first derivatives at 0, we get that |III|| x=0 ≤ C f C 0 δ for all δ ≤ δ 0 , with a constant C that only depends on the geometric data.For II, recall that ρ(x, y) = 1 cnδ n+1 η d(x,y) δ , where d is the Riemannian distance (induced by g); so for

ι
: M → N (double cover of M ) ι| M \ι −1 (B) (make a hole at B) push away from M keeping B fixed push away from M \ B close the hole at B continuously

Figure 1 :
Figure 1: Cut, deform, paste back in.The path of immersions in the second and third row reaches the same immersion depicted in the top-right picture.

( 0 ) 1 (
) to the function F ab defined to be constant on each of the two half-balls {z ∈ B n+1 1 (0) : z • nx < 0} and {z ∈ B n+1 0) : z • nx > 0}, with respective values a and b.This function must be the (distributional) gradient of d x .Indeed, for every

Proposition 3 . 2 .
Let N be a closed (n + 1)-dimensional Riemannian manifold with positive Ricci curvature and M a smooth minimal hypersurface as in Theorem 1.3.Denote by d M the distance function to M and by T ω = {x ∈ N : d M (x) < ω}, where ω < inj(N ).Then ∆d M ≤ 0 on T ω in the sense of distributions 4 .

Remark 5 . 1 .
This lemma uses n ≥ 2 to argue that {b} has codimension ≥ 2 (for n = 1 the lemma fails, e.g. for RP 1 ⊂ RP 2 ).Remark 5.2.By the construction of ρ in[6], ρ(x) = 0 when dist N (x, {b}∪(M \M )) > d δ for some d δ → 0 as δ → 0. This means that for δ sufficiently small the support of ρ has at least two(compact)  connected components one of which contains b (and thus D) while the union of the others contains an open neighbhourhood O 1 of M \M .Let O ⊂⊂ O 1 be an open set containing M \ M .For φ = 1 − ρ • ι, we have that the complement of supp φ has at least two (open) connected components in M , one containing D while the other contains ι −1 (O).Note that K = M \ ι −1 (O) is compact.These facts guarantee that φ vanishes in a neighbourhood of ∂ D and of ∂(ι −1 (O)) = ∂K, a condition that will be technically useful in Section 6.

Remark 5 . 3 (
choice of B).Choose the ball B in M to be concentric with D and with half the radius.Denote by R > 0 the radius of B. Let B = ι −1 (B): this is the union of two geodesic balls in M .The choices of B and φ will be kept until the end.
jump part of D(∇d M ) is characterized as the measure that is absolutely continuous with respect to H n S d M and with density that is given for H Remark 3.3.Sufficient regularity of S d M would actually imply that this jump part is 0 close to M , because of the smallness of M \ M (and the fact that S d M ∩ M = ∅).E.g. if S d M were a smooth hypersurface, then locally around a point in M \ M it would separate M into two stationary hypersurfaces (one on each side of S d M ) whose closures intersect only at M \ M , against the varifold maximum principle n -a.e.x ∈ S d M by (b − a) ⊗ nx (see e.g.[3, (3.90)]).Taking the trace and using (5) this implies: Lemma 3.2.Let ∆ denote the Laplace-Beltrami operator on T ω \ M .The singular (jump) part of ∆d M in T ω \ M is a negative measure (supported on S d M ).
and ∆d M is a negative Radon measure on this open set by Lemma 3.3, so that Tω