Uniqueness of Ricci flows from non‐atomic Radon measures on Riemann surfaces

In previous work (Topping and Yin, https://arxiv.org/abs/2107.14686), we established the existence of a Ricci flow starting with a Riemann surface coupled with a non‐atomic Radon measure as a conformal factor. In this paper, we prove uniqueness, settling Conjecture 1.3 of Topping and Yin (https://arxiv.org/abs/2107.14686). Combining these two works yields a canonical smoothing of such rough surfaces that also regularises their geometry at infinity.


Introduction
Suppose M is a two-dimensional smooth manifold with a conformal structure.By assumption our manifolds are connected but not supposed to be compact, unless otherwise specified.The conformal structure can be viewed as an equivalence class of Riemannian metrics where two metrics are deemed to be equivalent if they differ only by multiplication by a smooth function.The Ricci flow can be viewed as a way of evolving a representative g of the conformal structure under the PDE ∂g ∂t = −2K g g, where K g is the Gauss curvature of g.If we pick local isothermal coordinates x and y, and write the metric g = u(dx 2 + dy 2 ), then the conformal factor u evolves under the logarithmic fast diffusion equation Here, and throughout this paper, ∆ := ∂ 2 ∂x 2 + ∂ 2 ∂y 2 is the Laplacian with respect to the isothermal coordinates that we chose.
There is an extensive theory of Ricci flow in two dimensions that we briefly survey.In the case that the underlying surface M is closed, Hamilton and Chow [12,6] proved existence and uniqueness of smooth solutions g(t) on time intervals [0, T ) once a smooth initial metric g(0) has been prescribed, and the optimal existence time T was determined.A theory in which rough initial data was prescribed, instead of a smooth initial metric, was developed by Guedj and Zeriahi (existence [11]) and Di Nezza and Lu (uniqueness [7]).Their theory was again constrained to closed manifolds, but applied to more general Kähler Ricci flows.
The nature of the well-posedness question changes completely once the compactness of the underlying manifold is no longer assumed.The focus changes to the problem of how Ricci flow might lose information at spatial infinity, or be influenced by information coming in from infinity.Intuition from classical PDE theory can be misleading because the behaviour of solutions is heavily influenced by the fact that the solution and the underlying geometry are the same thing.
The smooth existence problem on general surfaces (without any assumption of compactness) was solved by Giesen and the first author [10], following the programme in [18].There it was proved that on a general smooth surface M , if we are given a smooth Riemannian metric g 0 , which is not assumed to be complete and is not assumed to have bounded curvature, there exists a smooth Ricci flow g(t) for t in an optimal time interval [0, T ), with g(0) = g 0 , and so that g(t) is complete for all t ∈ (0, T ).This instantaneously complete Ricci flow was maximally stretched in the sense that if g(t), t ∈ [0, T ), was any other smooth Ricci flow on M , with g(0) ≤ g(0) and with g(0) and g(0) in the same conformal class (but not assuming completeness of g(t)) then g(t) ≤ g(t) for t ∈ [0, T ∧ T ).
This existence theory was generalised to rough initial data by the present authors in [20].The essential idea there was that instead of prescribing the initial data as a smooth Riemannian surface, which could be viewed equivalently as a manifold with a conformal structure and a smooth function as a conformal factor, we replaced the smooth conformal factor by a Radon measure µ on M with the property that µ({x}) = 0 for all points x ∈ M .Once that relaxation had been made, it was necessary to weaken what it meant to satisfy the initial data.We say that a Ricci flow g(t), t ∈ (0, T ), attains µ as initial data (weakly) if the volume measures µ g(t) converge weakly to µ, written µ g(t) ⇀ µ.This means that for all ψ ∈ C 0 c (M ) we have ˆM ψdµ g(t) → ˆM ψdµ, as t ↓ 0.
(1.3)This is weak-* convergence when viewed in the dual of C 0 c (M ).In [20], we proved existence of such a Ricci flow on an optimal time interval that we will incorporate into our main theorem 1.1 below.This discussion addresses the existence problem in full generality, but says nothing about the problem of uniqueness that is so important in applications, except on closed manifolds as discussed above.In general dimension if one works in the class of smooth solutions and initial data that is complete and of uniformly bounded curvature then uniqueness was proved by Chen and Zhu [5], with a simplified proof given by Kotschwar [14].Once the hypothesis of bounded curvature is fully dropped, the problem becomes harder.In his influential paper [4], B.-L Chen proved that the stationary solution is the only smooth complete Ricci flow starting with Euclidean space in two or three dimensions.An audacious conjecture that is not currently generally accepted would be that any two complete Ricci flows g i (t), t ∈ [0, T i ), for i = 1, 2, on the same underlying manifold and with the same initial metric g 1 (0) = g 2 (0) must agree for t ∈ [0, T 1 ∧ T 2 ), in any dimension.This conjecture was proved by the first author for two-dimensional M in [19] but the question in general dimension is wide open.
In this paper we return to two-dimensional M and consider the uniqueness problem with initial data a nonatomic Radon measure µ attained as described in the weak sense (1.3) above.Even for smooth initial data attained in a weak sense the uniqueness problem is far from the smooth theory.As an illustration of this, in our previous work [20,Theorem 1.5] we proved nonuniqueness for Ricci flow starting with the Euclidean plane if the solution g(t), t ∈ (0, T ), is asked to attain the initial data in the sense that the Riemannian distance converges locally uniformly as t ↓ 0.
Nevertheless, in this paper we solve the problem for Radon measure µ initial data by proving the uniqueness part of the following main theorem, which incorporates the existence from [20].We use the notation M for the universal cover of M , and μ for the corresponding lift of µ to M .We write D for the unit disc in the plane, equipped with the standard conformal structure.
Theorem 1.1 (Main existence and uniqueness theorem).Let M be a two-dimensional smooth manifold equipped with a conformal structure, and let µ be a Radon measure on M that is nonatomic in the sense that Then there exists a smooth complete conformal Ricci flow g(t) on M , for t ∈ (0, T ), attaining µ as initial data in the sense that µ g(t) ⇀ µ as t ↓ 0 and so that if g(t), t ∈ (0, T ), is any other smooth complete conformal Ricci flow on M that attains µ as initial data in the same sense, then T ≤ T and for all t ∈ (0, T ).
Note that even in the case that µ is smooth, i.e. is the volume measure of a smooth conformal metric on M , this theorem has been open until this point, and is not contained in the uniqueness theorem of [19], because the initial data is only assumed to be attained by g(t) in the weak sense.
Observe that in the isolated cases in which T = 0, the existence part of the theorem above is vacuous but the uniqueness is not.We must still prove nonexistence of an alternative solution.
The solution constructed in Theorem 1.1 is the largest in the sense described in the following theorem.
Theorem 1.2 (Ordered initial data gives ordered Ricci flows).Let M , µ and T be as in Theorem 1.1, and let g(t) be the unique Ricci flow constructed in that theorem.Suppose now that ν is any other Radon measure on M with ν ≤ µ, and g(t), t ∈ (0, T ), is any smooth conformal Ricci flow on M (possibly incomplete) attaining ν as initial data.Then T ≤ T and for all t ∈ (0, T ).
This theorem gives the strongest assertion that the solution from Theorem 1.1 is 'maximally stretched' in the sense of [10].Note that even in the case that ν = µ and µ is smooth, this theorem has been open until this point, and is not contained in [10], again because the initial data is only assumed to be attained by g(t) in the weak sense.
The proof of Theorem 1.2, found in Section 3.2, will invoke the existence and uniqueness of Theorem 1.1.However, in order to prove Theorem 1.1 itself (Section 3.1) we will need to first prove a weaker 'maximally stretched' assertion that we give in Theorem 2.1, and prove in Section 2, in which we assume in particular that ν = µ.As mentioned earlier, the special case of M closed in Theorem 1.1 is covered in previous work.As a service to the reader, we translate the uniqueness proof of Di Nezza and Lu [7] into the language of this paper in Appendix C.
The theory in this paper has a number of immediate applications.It yields a complete analysis of expanding Ricci solitons in two dimensions, as will be explained in [15].This application requires crucially both the ability to work with very rough initial data and the ability to work on noncompact manifolds.Several of the arguments in [20] simplify significantly; some extensions will be discussed in [15].At a more general level, our work opens up the possibility of doing calculus on extremely rough surfaces by invoking the canonical smoothing that we are presenting here, and doing the analysis on the Ricci flow instead.
Acknowledgements: PT was supported by EPSRC grant EP/T019824/1.HY was supported by NSFC 11971451 and 2020YFA0713102.HY would like to thank Professor Yuxiang Li for a discussion on the result of Brezis-Merle [3].For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any author accepted manuscript version arising.

Construction of a maximally stretched solution from weak initial data
A central ingredient in the proof of Theorem 1.1 is the following construction of a maximally stretched solution starting with a nonatomic Radon measure µ on the unit disc D in the plane, or on the entire plane.
Theorem 2.1 (Existence of maximally stretched Ricci flow starting with µ).Let the Riemann surface M be either the disc or the plane, and let µ be a nonatomic Radon measure on M .If M is the disc we set T = ∞, while if M is the plane then we set Then there exists a smooth complete conformal Ricci flow g max (t) on M , for t ∈ (0, T ), attaining µ as initial data in the sense that µ gmax(t) ⇀ µ as t ↓ 0 and so that if g(t), t ∈ (0, T ), is any smooth conformal Ricci flow on M (not necessarily complete) that attains µ as initial data in the same sense, then T ≤ T and g max (t) ≥ g(t) for all t ∈ (0, T ).
In order to construct the maximally stretched solution of Theorem 2.1 we need to analyse a potential corresponding to the flow that would coincide with the standard potential of Kähler Ricci flow if we were working on a closed surface.
Understanding the right way of defining and localising the potential in the noncompact case is a key part of our task in this paper.
Lemma 2.2 (Construction of a potential).Suppose the Riemann surface M is either the disc or the plane.Suppose we have a smooth complete conformal Ricci flow g(t) on M , t ∈ (0, T ), for T ∈ (0, ∞], with conformal factor u : M ×(0, T ) → (0, ∞), taking a Radon measure µ on M as initial data.Then there exists a smooth potential function ϕ : M × (0, T ) → R satisfying for each t ∈ (0, T ) and ∂ϕ ∂t = log ∆ϕ (2.2) throughout M × (0, T ).Moreover, there exists a function weakly, and for all x ∈ M .
By virtue of (2.1) and (2.2), the potential ϕ and its initial values ϕ 0 are only determined up to the addition of a time-independent harmonic function on M .
To avoid repetition in Appendix C, we extract a portion of the proof of Lemma 2.2 in the following.
In the final part we are also told that ϕ(t) is subharmonic, and in particular ϕ(x, t) ≤ for all x ∈ B R and small enough r > 0 so that B r (x) ⊂⊂ B R .We can pass this inequality to the limit t ↓ 0 to give ϕ 0 dx, and hence ϕ 0 dx.
On the other hand, we have established that ϕ 0 is upper semi-continuous, and so Combining these two inequalities gives Part (3).
Remark 2.4.Classical potential theory gives other ways of phrasing the proof, some of which give additional information that we do not require.For example, the integrals in (2.7) are increasing in r.See, for example, [2, Section 3].
The following remark will be used in the proof of Lemma 2.2 and later also.
Remark 2.5.Whenever we have a smooth complete Ricci flow g(t) on a surface M , t ∈ (0, T ), and we write it g(t) = u(t)g 0 for conformal factor u and fixed background metric g 0 , then for each x ∈ M , The reason is that the Ricci flow equation (1.1) implies that while because of the completeness, B.-L. Chen's scalar curvature estimate [4] tells us that the Gauss curvature satisfies the lower bound K g(t) ≥ − 1 2t for all t ∈ (0, T ).Then ∂ ∂t Proof of Lemma 2.2.Define τ := T 2 ∧ 1 and fix a smooth solution ϕ τ to the equation ∆ϕ τ = u(τ ) on M .Such a solution can be found using Lemma A.1 and is unique up to the addition of a harmonic function.We define the required smooth potential By applying ∆ and using the Ricci flow equation (1.2) we obtain at each point and so ϕ satisfies (2.1).Moreover, combining this with the time derivative of (2.8) tells us that ϕ satisfies (2.2). Pick We have established the first hypothesis of Lemma 2.3.
Next, fix any R > R still with B R ⊂⊂ M .Then for any 0 < s ≤ t ≤ τ , we can appeal to [20,Lemma 3.2] to find that for some η > 0 depending only on R/R.In particular, by taking a limsup as s ↓ 0 we obtain where L < ∞ is independent of t (keeping in mind that t ≤ τ ≤ 1).This implies hypothesis (2.5) of Lemma 2.3.
Applying Lemma 2.3 then gives ϕ 0 and all its desired properties except for the equation (2.3).By (2.1), for every ξ ∈ C 2 c (M ) we have and by taking a limit t ↓ 0 we obtain i.e. ϕ 0 is a weak solution of ∆ϕ 0 = µ as required.
In practice we must contemplate other solutions ψ to the potential function evolution equation (2.2) with the same initial data ϕ 0 that was induced in Lemma 2.2.It will be important to verify that such ψ cannot jump up unreasonably from ϕ 0 as t lifts off from zero, and that is the content of the following lemma.
Lemma 2.6 (Space-time upper semi-continuity).Suppose the Riemann surface M is either the disc or the plane.Suppose, for T > 0, that we have a smooth function throughout, and so that as t ↓ 0, where ϕ 0 ∈ L 1 loc (M ) admits a representative for which ϕ 0 dx for all x ∈ M .Then for all x ∈ M , all x n → x and all t n > 0 with t n ↓ 0, we have Proof.By subharmonicity of ψ(t n ), the mean value inequality gives for sufficiently small r > 0 and all n.Therefore lim sup where the final equality here follows from the L 1 loc (M ) convergence hypothesis (2.12).We can then take a limit r ↓ 0 to conclude.
We will prove Theorem 2.1 using the following lemma.Note that in this lemma, the flow g max (t) depends a priori on g(t), although ultimately this dependency will be lifted.

Lemma 2.7 (Maximal potential function construction).
Let the Riemann surface M be either the disc or the plane, and let µ be a nonatomic Radon measure on M .If M is the disc we set T = ∞, while if M is the plane then we set is a smooth conformal Ricci flow on M , for t ∈ (0, T ), T > 0, that attains µ as initial data in the sense that µ g(t) ⇀ µ as t ↓ 0. Then T ≤ T and there exists a smooth complete conformal Ricci flow g max (t) on M , for t ∈ (0, T ) that also attains µ as initial data, and which satisfies g max (t) ≥ g(t) for all t ∈ (0, T ).
Let ϕ ∈ C ∞ (M × (0, T )) be a potential coming from Lemma 2.2, together with the corresponding ϕ 0 , corresponding to g max (t). ) Proof of Lemma 2.7.The only case in which T = 0 is the case that M is the plane and µ is the trivial measure.Our sole task in this case is to show the nonexistence of any smooth conformal Ricci flow g(t) on M , for t ∈ (0, T ), T > 0, that attains the trivial measure µ as initial data.Suppose such a flow exists.Then for each R > R > 0, and 0 where the proof establishes that η is the volume of B R with respect to the complete conformal hyperbolic metric on B R. Sending s ↓ 0 gives , which is a contradiction.Thus from now on we may assume that T > 0, though µ could still be the trivial measure on the disc.
As a warm-up to the techniques involved in constructing g max (t) we first reduce the lemma to the case that the given Ricci flow g(t) is complete, and T = T .This will be possible by replacing g(t) by the flow ĝ(t) arising in the following claim.
Proof of Claim 1: For each δ ∈ (0, T ), we take g(δ) as smooth (possibly incomplete) initial data for a Ricci flow on M .The previous two-dimensional theory [10,19] tells us that there exists a unique Ricci flow g δ (t), t ∈ [δ, T δ ) (for some T δ ∈ (δ, ∞]) that is complete for t ∈ (δ, T δ ), and for which g δ (δ) = g(δ).Moreover, the earlier theory tells us that if T δ < ∞ then µ g δ (t) (M ) → 0 as t ↑ T δ , and that g δ (t) ≥ g(t) at times at which both flows exist because g δ (t) is maximally stretched [10], i.e., it lies above any other smooth Ricci flow with the same, or lower, initial data.We conclude that g δ (t) must exist at least for t ∈ [δ, T ).
If we pick a sequence δ i ↓ 0, then t → g δ i (t + δ i ) gives a sequence of Ricci flows whose initial data converges weakly to µ.We can then apply [20,Lemma 4.2] to deduce that after passing to a subsequence in i, there exists T 0 ∈ (0, T ) such that the flows converge smoothly locally on M × (0, T 0 ) to a new Ricci flow ĝ(t) on M , for t ∈ (0, T 0 ), that has µ as its initial data (as for g(t)) but that is now complete.
Translating the ith flow by δ i in time, we find that the Ricci flows g δ i (t) also converge smoothly locally on M × (0, T 0 ) to ĝ(t).Because g δ i (t) ≥ g(t), we also have ĝ(t) ≥ g(t) on M × (0, T 0 ).We would now like to extend ĝ(t) in time.A basic consequence of ĝ(t) attaining µ as initial data is that We can then extend ĝ(t) by taking the unique complete Ricci flow starting with ĝ(η), for small η ∈ (0, T 0 ), using [10,19].This extension exists for all time if M is the disc, while if M is the plane then it exists for a time as η ↓ 0. Indeed, in this case the extension will satisfy as t ↑ T .Because ĝ(t) is maximally stretched, we can extend the inequality ĝ(t) ≥ g(t) to M × (0, T ∧ T ).Then g(t) cannot exist beyond time T since it would have no volume left, and so T ≤ T .End of proof of Claim 1.
As a result, we may replace g(t), for t ∈ (0, T ), by the complete flow ĝ(t) for t ∈ (0, T ), as claimed before Claim 1.We write u(t) for the conformal factor of what we are now calling g(t).
We now turn to constructing the g max (t) required in the lemma as the limit of flows defined on smaller domains.At this stage of the argument we are unsure whether or not the Ricci flow we have just constructed in Claim 1 (now renamed g(t)) can serve as the required g max (t).It will turn out to be the same, but this fact is not clear at this stage.
In the following, we set Claim 2: For R ∈ (0, R * ), there exists a smooth conformal complete Ricci flow g R (t) on B R , t ∈ (0, ∞), with conformal factor u R (t), such that µ g R (t) ⇀ µ weakly as t ↓ 0, where µ is restricted to B R , and with the monotonicity property that if and so that Proof of Claim 2: For each R ∈ (0, R * ), and each δ ∈ (0, T ), take the unique instantaneously complete Ricci flow on B R , for t ∈ [δ, ∞), with conformal factor denoted by u R,δ satisfying u R,δ (δ) = u(δ) on B R , where we recall that u(t) is the conformal factor of g(t).
Certainly u R,δ (t) ≥ u(t) for all t ∈ [δ, T ) because an instantaneously complete solution (with smooth initial metric) such as that corresponding to u R,δ (t) is always larger than any other solution with the same initial data as a result of the maximally stretched property [10].Similarly, for 0 We can now mimic the proof of Claim 1 and appeal to [20, Lemma 4.2], with M there equal to B R here, to find that there exists ) on B R whose volume measures converge weakly to (the restriction of) µ as t ↓ 0. By construction, the monotonicity (2.15 Because of the phrasing of [20,Lemma 4.2], this only gives the compactness on some time interval (0, T R ).However, if we can verify that we have local positive lower and upper bounds for u R,δ (t) on B R × (T R /2, ∞) that are uniform as δ ↓ 0 then we can invoke parabolic regularity theory and pass to a subsequence to get smooth local convergence of u R,δ i (t) to an extended u R : B R × (0, ∞) → (0, ∞).Indeed, by invoking Yau's Schwarz lemma and B.-L. Chen's scalar curvature estimate K g(t) ≥ − 1 2t from [4], as originating in [9], we have uniform lower bounds u R,δ (t) ≥ 2(t − δ)h R , where h R is the conformal factor of the unique complete conformal hyperbolic metric on B R , that are valid for all t ≥ δ.Moreover, for any r ∈ (0, R) if we pick sufficiently large C r < ∞ so that u R (T R /2) < C r h r on B r , then also for δ ∈ (0, T R /2) we have u R,δ (T R /2) ≤ u R (T R /2) < C r h r on B r , and these upper bounds propagate forwards to give u R,δ (t + T R /2) ≤ (C r + 2t)h r on B r , by the maximally stretched property of the Ricci flow t → (C r + 2t)h r .These lower and upper bounds on the conformal factors u R,δ (t) suffice to give compactness on B R × (0, ∞).
Moreover, these limit flows u R (t) inherit the ordering that u R (t) ≥ u(t) on B R for all t ∈ (0, T ), and also that if 0 End of proof of Claim 2.
If we restrict our attention to an arbitrary compact set K in M , the inequalities for the conformal factors from Claim 2 are enough to obtain positive lower and upper bounds on the conformal factors u R on K that are uniform in R as R ↑ R * .
Thus we can find a sequence R i ↑ R * so that u R i (t) converges smoothly locally on M × (0, T ) to a limit u max (t), which is the conformal factor of a Ricci flow g max (t).By construction, we have u(t) ≤ u max (t) on M for all t ∈ (0, T ).One consequence is that g max (t) inherits the completeness from g(t).We also have, for all R ∈ (0, R * ), that u max (t) ≤ u R (t) throughout B R × (0, T ).
Furthermore, because u max (t) is sandwiched between u(t) and u R (t), both of which weakly attain µ as initial data, we see that u max (t) does also.
We have constructed the Ricci flow g max (t) that is required in the lemma, and this induces the potential ϕ (with initial data ϕ 0 ) by Lemma 2.2.It remains to investigate how ϕ relates to the potential ψ.In order to do this, we will relate ϕ to approximating potentials on subdomains of M .For each R ∈ (0, R * ), we feed the Ricci flows with conformal factors u R into Lemma 2.2, with M there equal to the ball B R here, and T there equal to ∞ here, to yield a potential throughout, and with initial data (ϕ R ) 0 satisfying ∆(ϕ R ) 0 = µ weakly.Since ϕ 0 − (ϕ R ) 0 is weakly harmonic on B R , by redefining ϕ R by adding a time-independent harmonic function we may assume that (ϕ R ) 0 agrees with ϕ 0 .We emphasise that by (2.4), not only do we have ϕ R (t) → ϕ 0 in L 1 loc (B R ) as t ↓ 0, but also the convergence holds pointwise.
The potentials ϕ R inherit the monotonicity of u R with respect to R. Indeed, for 0 for every t > 0, and then by continuity, for every point x ∈ B R 1 and every t > 0 we have throughout B R × (0, T ).This inequality will have multiple applications.The first is that we can combine with the monotonicity of ϕ R with respect to R in order to define φ : and to deduce that φ ≥ ϕ. (2.17) Moreover, the function φ will be smooth.To see this, pick an arbitrary compact set K ⊂ M × (0, T ) and choose R ∈ (0, R * ) sufficiently large so that K ⊂ B R × (0, T ).For R ∈ [ R, R * ), the potentials ϕ R , restricted to B R × (0, T ), will be sandwiched between ϕ and ϕ R.Meanwhile, we already showed that u R i (t) converges smoothly locally on M × (0, T ).Elliptic regularity theory applied to the equation ∆ϕ for sufficiently large i, gives control on all spatial derivatives of ϕ R i (t) over K that is uniform as i → ∞.By appealing to the equation we then obtain C k space-time control of ϕ R i over K that is uniform in i.By passing to a further subsequence we see that ϕ R i converges smoothly locally, and thus its pointwise limit φ is smooth.
We claim now that we have equality in the inequality (2.17).Because we have established that φ is smooth, it suffices to prove the equality at points x ∈ M where ϕ 0 > −∞ and at times t ∈ (0, T ).Choose R > |x| so that x ∈ B R. For arbitrary ε > 0, we can pick δ ∈ (0, t) sufficiently small so that ) and below ϕ R(x, δ), and so |ϕ R (x, δ) − ϕ 0 (x)| < ε/3 also.Meanwhile, and by the smooth local convergence of u R i to u max , for sufficiently large i we have In particular, for sufficiently large i also to ensure that R i ≥ R, we have (2.18) Because ε > 0 was arbitrary, taking the limit i → ∞ forces as required.
The next step is to show that as R increases, making ϕ R (x, t) decrease, these potentials will remain above the globally-defined potential ψ considered in the statement of the lemma, once restricted to B R .
Claim 3: For every R ∈ (0, R * ), we have Notice that if this claim is true then we can apply it for R = R i and let i → ∞ to obtain that for every x ∈ M and t ∈ (0, T ) we have which will complete the proof of Lemma 2.7.
Proof of Claim 3: It suffices to prove the claim over B R × (0, T 1 ) for arbitrary T 1 ∈ (0, T ); we fix such a T 1 .
Since u R is the conformal factor of a complete Ricci flow on B R , t > 0, we can appeal again to the idea from [9] and use Yau's Schwarz lemma to deduce that u R (t) ≥ 2th R on B R , for every t > 0, where h R is the conformal factor of the unique complete conformal hyperbolic metric on B R .This lower bound for u R gives us lower control on ϕ R because for every x ∈ B R and every t ≥ τ > 0, we have and so (2.19) gives both control on how ϕ R (t) blows up near the boundary of B R and also on how fast ϕ R (t) can decrease in time.For the latter, for points where ).As a consequence, for each δ > 0 we can define a modified potential ϕ R,δ (t) := ϕ R (t + δ) + F (δ) + δ(1 + t) on B R ×(0, ∞) so that ϕ R,δ (0) is a smooth function on B R that is bounded below, and so that ϕ R,δ (0) ≥ ϕ 0 + δ.
(2.22) Subclaim: For δ > 0 chosen sufficiently small so that T 1 + δ < T , we have Observe that if we can prove this subclaim, then by taking a limit δ ↓ 0 we will have proved Claim 3, and hence the whole lemma.
At a heuristic level, (2.23) is true at t = 0 by (2.22) because ψ takes ϕ 0 as initial data.However, ψ only admits initial data as a L 1 loc limit, so this statement requires care.
We will establish the subclaim using a classical maximum principle approach.In lieu of an ordering like (2.23) on the boundary ∂B R , we will need to show that ϕ R,δ − ψ blows up in an appropriately uniform sense near the boundary and we will achieve this by proving an estimate of the form There are three ingredients for this.The first is that the potential ψ has a finite upper bound L over B R × (0, T 1 ].If this were not the case then we would be able to find a sequence x n ∈ B R , and times t n ↓ 0, with ψ(x n , t n ) → ∞.By passing to a subsequence, we could assume that x n → x for some x ∈ B R .But Lemma 2.6 would tell us that lim sup giving a contradiction.We conclude that The second ingredient for (2.24) is that ϕ has a finite lower bound L over B R × [δ/2, T 1 + δ].This is simply by continuity of ϕ and compactness of the spacetime region considered.An immediate consequence (see (2.16)) is that the larger function ϕ R has the same lower bound over the part of the region considered where it is defined, i.e.
The third and final ingredient is (2.19) with τ = δ/2.This implies that throughout B R and for t ∈ [δ/2, where C 0 < ∞ depends on L, T 1 , R and δ.In particular, for t ∈ (0, where C 1 < ∞ depends on L, L, T 1 , R and δ.This completes the proof of (2.24).
We now complete the proof of the subclaim (and hence the whole lemma) by contradiction.If it fails for some valid δ then we can find a point in B R × (0, T ) where ψ ≥ ϕ R,δ .We claim that then we can find a point (x 0 , t 0 ) ∈ B R × (0, T ) so that ψ(x 0 , t 0 ) = ϕ R,δ (x 0 , t 0 ) but so that for all t ∈ (0, t 0 ) we have ψ(t) < ϕ R,δ (t) throughout B R .To find this point (x 0 , t 0 ), first define We can then take sequences t n ↓ t 0 and x n ∈ B R with After passing to a subsequence we may assume that x n → x 0 ∈ B R .Note that x 0 cannot lie on ∂B R because of the control near the boundary given by (2.24).
If t 0 = 0, then we appeal to Lemma 2.6.This gives us the final inequality of which contradicts (2.22).Thus t 0 > 0 and we find that But by definition of t 0 , for every t ∈ (0, t 0 ) we have ψ(t) < ϕ R,δ (t), so and we must have equality throughout (2.27), and we have found the required point (x 0 , t 0 ).
It remains to compute derivatives at this point (x 0 , t 0 ) in order to generate a contradiction.Applying the second derivative test to ϕ R,δ (t 0 ) − ψ(t 0 ) at x 0 , where it achieves its minimum value of zero, we find that at (x 0 , t 0 ).By the characterisation of t 0 , we must have Evaluating at (x 0 , t 0 ), subtracting and applying (2.29) gives which contradicts (2.30).
We can derive Theorem 2.1 rapidly from Lemma 2.7.
Proof of Theorem 2.1.As a preliminary step, we consider the case that µ is the trivial measure and M is the plane.In this case T = 0 so the existence of g max (t) is a vacuous statement.Moreover, it is a consequence of Lemma 2.7 that no Ricci flow g(t) as in the theorem can exist because T > 0 is impossible.This completes the proof in that case.
In all remaining cases, we have T > 0. We start by picking a smooth complete conformal Ricci flow g(t) on M for t ∈ (0, T ) with µ as initial data.In the case that M is the disc and µ is the trivial measure we simply take g(t) to be the big-bang Ricci flow with conformal factor 2th, where h = h 1 (as in (2.20)) is the conformal factor of the Poincaré metric.
In the remaining cases in which µ is not the trivial measure, we can use our previous existence result [20,Theorem 1.2] to find a smooth complete conformal Ricci flow g(t) for t ∈ (0, T ) such that µ g(t) ⇀ µ as t ↓ 0. This means that we can apply Lemma 2.7, even for T = T .The output of the lemma is a new Ricci flow g max (t) ≥ g(t) for t ∈ (0, T ) with µ as initial data, and also a potential ϕ with the properties described in the lemma, and with t ↓ 0 limit ϕ 0 .We denote the conformal factor of g max (t) by u(t).
We claim that this g max (t) serves as the g max (t) required in Theorem 2.1.For this to be true we must establish that for every g(t), t ∈ (0, T ) as in the theorem, we have T ≤ T and g max (t) ≥ g(t) for all t ∈ (0, T ).To see this we will make a second application of Lemma 2.7, this time with T there equal to T here, but with g(t) there equal to g(t) here.The output of the lemma is a flow that we call gmax (t), t ∈ (0, T ), to distinguish it from the g max (t) already introduced in this proof.Similarly, we call the new potential φ, its t ↓ 0 limit φ0 , and the new conformal factor ũ(t).
We would like to exploit our knowledge of the potentials ϕ and φ to prove that, in fact, the two Ricci flows g max (t) and gmax (t) coincide.If we achieve that then we will have proved that for all t ∈ (0, T ), as required.
To establish this equality we first consider the modified potential Note that the static function ϕ 0 − φ0 is harmonic (not just a weakly harmonic function that is an almost-everywhere representative of a smooth harmonic function) by the characterisations of ϕ 0 and φ0 given by (2.13) of Lemma 2.2.Consequently ψ satisfies the potential function evolution equation (2.13).It also satisfies ψ(t) → ϕ 0 in L 1 loc (M ) as t ↓ 0, so the first application of Lemma 2.7 tells us that ψ ≤ ϕ throughout M × (0, T ), i.e. φ(t) − ϕ(t) ≤ φ0 − ϕ 0 . (2.31) On the other hand, we could define a modified potential ψ(t) := ϕ(t) − ϕ 0 + φ0 , which would again satisfy (2.13), and this time satisfy ψ(t) → φ0 in L 1 loc (M ) as t ↓ 0. We can then use the information from the second application of Lemma 2.7 to obtain that ψ ≤ φ throughout M × (0, T ), i.e. (2.32) Comparing (2.31) and (2.32) we find that We deduce that the conformal factors of g max (t) and gmax (t) are equal so the flows themselves must coincide as required.

Proofs of the main theorems
In this section, we prove Theorem 1.1 and Theorem 1.2.
3.1 Proof of the main well-posedness theorem 1.1 Because we are claiming both existence and uniqueness, we may as well lift to the universal cover.In particular, the uniqueness assertion on the universal cover, once proved, will tell us that the flow will descend to the original manifold.We thus reduce to the three cases that M is S 2 , the disc D or the plane.
If M is S 2 then our task is simplified because of the compactness of S 2 .Indeed, on closed manifolds the existence and uniqueness statements that imply this case have already been proved in the more general Kähler setting by Guedj and Zeriahi [11] and Di Nezza and Lu [7].For convenience, we translate the proof of uniqueness on S 2 into the language of this paper in Appendix C.
Suppose then that we are in one of the remaining cases that M is the disc or the plane.If T = 0 then M is the plane and µ is trivial, and then both the uniqueness statement and the vacuous existence statement of Theorem 1.1 are already included in Theorem 2.1.Therefore we may assume that T > 0 from now on.We obtain the existence of a maximally stretched solution g max (t), for the required length of time, from Theorem 2.1.We claim that this will serve as the g(t) required in Theorem 1.1.To establish this it remains to prove uniqueness in these cases.
For the disc case, let g(t) be the other solution in the assumptions of Theorem 1.1.Since both g(t) and g(t) take the same measure µ as the initial data, we know that for every η ∈ C 0 c (D), For R ∈ (0, 1), by picking any η ∈ C 0 c (D, [0, 1]) with η ≡ 1 on B R and using the fact that g(t) ≥ g(t) (by Theorem 2.1), we have For any t > 0 for which both g(t) and g(t) are defined, we apply Lemma 3.3 of [19] to the time interval [ε, t] to find that for 1 2 < r 0 < r . By (3.1), taking ε ↓ 0 first and then R ↑ 1 in the above inequality, we obtain for any r 0 ∈ (1/2, 1).Since g(t) ≥ g(t), we deduce that g(t) = g(t) as long as both of them are defined.
For the R 2 case, using the same notation as above, for any R > 0 we still have (3.1).With the standard flat metric on R 2 as background, denote the conformal factors of g(t) and g(t) by u and ũ respectively.Notice that by the maximality in Theorem 2.1, ũ(t) ≤ u(t).
It is our goal to show u(τ ) = ũ(τ ) for each τ > 0 for which both g and g are defined.For that purpose, we need the following lemma, which is close in spirit to [16,Theorem 2.1], and closer still to the proof of the variant to be found in [8,Theorem 4.4.1].Lemma 3.1.Suppose g(t) ≥ g(t) are (conformal) Ricci flows on R 2 for t ∈ (0, τ ], with conformal factors u(t) and ũ(t) respectively, such that for some ε > 0 we have ũ(x, t) ≥ εt (|x| log |x|) 2  (3.2) for every x ∈ R 2 \ B 2 and every t ∈ (0, τ ].Then for 0 < s < t ≤ τ , R > 2 and any m ∈ (0, 1) we have for some constant c 0 depending only on m.
The proof of this lemma is almost the same as that of Theorem 2.1 of [16], except that we have assumed that (3.2) holds uniformly on (0, τ ] so that the inequality (3.3) is valid for arbitrarily small s.Moreover, since we have assumed u ≥ ũ, we do not need to take the positive part as in [16].

Proof of the ordering theorem 1.2
In this section we prove Theorem 1.2.Our first observation is that given the uniqueness result in Theorem 1.1, we may as well assume that M is simply connected.If otherwise, we may consider the lift to the universal cover M and the uniqueness implies that the lift of g(t) is the unique Ricci flow starting from the lift of µ.If we know it is larger than the lift of g, then g is larger than g as we desire.
Secondly, we may assume that g is complete.If M is compact, then this is automatic.If M is the disc or the plane, we apply Theorem 2.1 to get a complete Ricci flow gmax (t) with ν as the initial data and satisfying gmax (t) ≥ g as long as they both exist.Note that ν is forced to be nonatomic, as required by Theorem 2.1, because ν ≤ µ.By replacing g(t) with gmax (t), we may assume that g(t) is complete as desired.
Recall that for the proof of existence in [20], we constructed a sequence of smooth initial metrics g i converging to the Radon measure µ in some sense (see Lemma 4.1 of [20]).If g i (t) is the complete Ricci flow from g i (Theorem 1.6 of [20]), we proved that they converge to a complete Ricci flow starting from µ.A priori, this limit flow may depend on the choice of approximations.
However, as a corollary of the uniqueness result in Theorem 1.1, we may now take any approximating sequence g i and the argument in [20] gives the unique complete Ricci flow solution starting with the given initial measure.For the purpose of proving Theorem 1.2, it therefore suffices to find two sequences of approximating smooth initial metrics for µ and ν respectively such that the order is preserved, because their limits, which by the uniqueness result are g(t) and g(t), will be ordered.
The following mollification result is similar to [20,Lemma 4.1].It is slightly simpler since we can now work on the universal cover.On the other hand we claim a little more in the sense that we need to compare the smoothings of two measures that are known to be ordered.
Lemma 3.2 (Smoothing lemma).Suppose the Riemann surface M is either the disc, the plane or the sphere.We can find a map from the space of Radon measures µ on M and numbers h > 0 to the space of conformal Riemannian metrics g h (µ) on M of finite total volume, with the properties that for every fixed Radon measure ν, we have as h ↓ 0, and so that for all Radon measures ν 1 and ν 2 on M with ν 1 ≤ ν 2 , and for every h > 0, we have g h (ν 1 ) ≤ g h (ν 2 ).
Proof.Consider first the case that M is the plane.Let g 0 be a metric on M corresponding to the round punctured unit sphere, pulled back by stereographic projection.Given a Radon measure ν on the plane, and h > 0, we can restrict ν to the ball B 1/h and then mollify ν in the traditional way over the whole plane (with respect to the scale parameter h) to give a smooth conformal factor of a degenerate Riemannian metric on R 2 , with compact support.If we add on hg 0 then we obtain a nondegenerate Riemannian metric g h (ν) with the required properties.
The case that M is the disc can be handled in a similar manner.Given a Radon measure ν on D and h ∈ (0, 1 2 ), we restrict to B 1−2h , mollify and add hg 1 , where g 1 is (say) the Lebesgue measure of the unit disc.
The case that M = S 2 will follow instantly from mollification alone, although we should use mollification with respect to the distance of the round spherical metric.
The proof below is adapted from Section 9 and 10 in [13].It was shown there that we can use the method of upper and lower solutions to solve where c is a constant and h is a smooth function.Notice when c = −1 and h = −e −f , (B.2) becomes (B.1).
Lemma B.2 (Lemma 9.3 in [13]).Let c < 0 be a constant and h ∈ C ∞ (S 2 ) be a smooth function.If there exist smooth upper and lower solutions u + and u − in the sense that and if u − ≤ u + , then there is a smooth solution u to (B.2) satisfying u − ≤ u ≤ u + .
Lemma B.1 follows from the above lemma if we can construct some upper and lower solutions that are bounded by a constant depending only on p and an upper bound for the L p norm of h.The following construction is from [13].However, it is simplified since we have assumed that f ≤ 0, i.e. h ≤ −1.
Lemma B.3 (From Lemma 9.5 and Theorem 10.5 (a) in [13]).For the case c = −1 and h ≤ −1, we can construct smooth functions u + and u − satisfying for some C depending only on p and an upper bound for h L p .
Proof.Let h be the average of h with respect to g 0 , which by our assumption is no larger than −1.In the following proof, the constant C varies from line to line and depends only on p and an upper bound for h L p .Solve with the normalization that the average of v vanishes, i.e. v = 0. Hence, the usual L p estimate implies that Set u + = v + v C 0 and we find that We check that So far we have constructed an upper solution u + and a lower solution u − , however, it is not clear if u − ≤ u + .Since u + and u − are both bounded by C, we have The key observation is that since h < 0, u − − 2C is also a lower solution.
C Uniqueness on the 2-sphere In this section, we give a self-contained proof of the uniqueness part of Theorem 1.1 in the case that the universal cover M is S 2 .Some aspects of the proof are easier than in the proof of Theorem 1.1 owing to the compactness of S 2 .In particular, it is straightforward to directly construct a potential corresponding to any Ricci flow.The arguments used here are adapted from [7].
The proof consists of two parts.The first part proves the existence of the potential flow and translates the uniqueness problem into that of the potential flow.The second part uses various maximum principles to prove the uniqueness.
By passing to the universal cover, we may assume that M = S for any continuous function η on S 2 .By setting η ≡ 1, we thus deduce that µ g(t) (S 2 ) = µ(S 2 ) − 8πt.In particular, in the case that µ is the trivial measure then not only is the existence trivial (because T = 0) but also, no other solution can exist, giving uniqueness.We may therefore assume that µ is not trivial, and T > 0. By scaling, we may also assume that µ(S 2 ) = 4π.
The existence of at least one solution for the required time is known from [20], cf.[11].
Recall that g 0 is the round metric of curvature 1 and ∆ 0 is the Laplacian of g 0 .If we let u(t) be the conformal factor of g(t) with respect to g 0 , then the Ricci flow equation (1.1) becomes Since g(t) attains the measure µ as initial data, we have the following estimates for u(t): (i) As justified above, the area of g(t) is given by This follows from the monotonicity of u t given in Remark 2.5.

C.1 The flow of the potential
The main result of this subsection is the existence of a potential flow.
Lemma C.1.Given µ, g(t) and u(t) as above, there exists a smooth function ϕ : The pointwise limit lim t↓0 ϕ, denoted by ϕ 0 , exists in R ∪ {−∞}.Moreover, ϕ 0 is an upper semi-continuous function satisfying in the sense of distributions.
(5) For any x ∈ S 2 , we have It is possible that both sides of the above equation are minus infinity.
(7) For any p > 1, there exist δ > 0 and C > 0 such that The potential ϕ is unique up to the addition of a time-independent constant.To see this, we take any continuous function η and compute For the proof of Part (6), we apply Lemma 2.6 to φ and notice that the assumptions there are proved by our previous application of Lemma 2.3.
For Part (7), we first find a cover of S 2 by balls B  Here B j (m) is the ball of the same center as B j , but m times the radius.By Lemma 3.2 of [20], there is δ > 0 such that for any t ∈ (0, δ), ˆBj u(t)dµ g 0 ≤ 2π p .
We may assume that the radius of B j are small so that ˆBj (2) |∆ 0 ϕ(t)| dµ g 0 ≤ 3π p .
Adding these together, we get Part (7).
Our goal is to show that ϕ 1 (t) ≡ ϕ 2 (t) since that will give our desired conclusion that u 1 ≡ u 2 .
Since ϕ i is smooth for t > 0, we have ∂ t ϕ i (x 0 , t 0 ) ≥ 0 and ∆ 0 ϕ i (x 0 , t 0 ) ≤ 0, which is a contradiction to the equation If both u 1 and u 2 are smooth on S 2 × [0, 1/2), then by the well-known uniqueness of smooth Ricci flow on compact manifolds, we always have u 1 = u 2 .Or, one may apply the classical maximum principle to (C.6) to see that ϕ 1 = ϕ 2 on S 2 × [0, 1/2).
When the initial data is only a measure, we know from Lemma C.1 that ϕ i (0) may not be bounded from below so that the classical maximum principle argument will not work.We use the same argument as in Di Nezza and Lu to solve this problem.

(C.14)
If the claim is not true, then we have a sequence of (y i , s i ) satisfying y i → y 0 , s i ↓ 0 and ξ(y i , s i ) + δ(e s i − 1) ≤ ψ 2 (y i , s i ).
By the smoothness of ξ at t = 0 and the upper semi-continuity of ψ 2 (a property inherited from ϕ 2 ), we take the limit i → ∞ in the above inequality to get a contradiction to (C.16), which proves the claim.