A variant prescribed curvature flow on closed surfaces with negative Euler characteristic

On a closed Riemannian surface (M,g¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(M,{\bar{g}})$$\end{document} with negative Euler characteristic, we study the problem of finding conformal metrics with prescribed volume A>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A>0$$\end{document} and the property that their Gauss curvatures fλ=f+λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\lambda = f + \lambda $$\end{document} are given as the sum of a prescribed function f∈C∞(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in C^\infty (M)$$\end{document} and an additive constant λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}. Our main tool in this study is a new variant of the prescribed Gauss curvature flow, for which we establish local well-posedness and global compactness results. In contrast to previous work, our approach does not require any sign conditions on f. Moreover, we exhibit conditions under which the function fλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\lambda $$\end{document} is sign changing and the standard prescribed Gauss curvature flow is not applicable.


Introduction
Let (M, ḡ) be a two-dimensional, smooth, closed, connected, oriented Riemann manifold endowed with a smooth background metric ḡ.A classical problem raised by Kazdan and Warner in [11] and [10] is the question which smooth functions f : M → R arise as the Gauss curvature K g of a conformal metric g(x) = e 2u(x) ḡ(x) on M and to characterise the set of all such metrics.
For a constant function f , this prescribed Gauss curvature problem is exactly the statement of the Uniformisation Theorem (see e.g.[16], [12]): There exists a metric g which is pointwise conformal to ḡ and has constant Gauss curvature K g ≡ K ∈ R. We now use this statement to assume in the following without loss of generality that the background metric ḡ itself has constant Gauss curvature K ḡ ≡ K ∈ R. Furthermore we can normalise the volume of (M, ḡ) to one.We recall that the Gauss curvature of a conformal metric g(x) = e 2u(x) ḡ(x) on M is given by the Gauss equation K g (x) = e −2u(x) (−∆ ḡ u(x) + K). (1.1) Therefore the problem reduces to the question for which functions f there exists a conformal factor u solving the equation Given a solution u, we may integrate (1.2) with respect to the measure µ ḡ on M induced by the Riemannian volume form.Using the Gauss-Bonnet Theorem, we then obtain the identity where dµ g (x) = e 2u(x) dµ ḡ (x) is the element of area in the metric g(x) = e 2u(x) ḡ(x).We note that (1.3) immediately yields necessary conditions on f for the solvability of the prescribed Gauss curvature problem.In particular, if ±χ(M ) > 0, then ±f must be positive somewhere.Moreover, if χ(M ) = 0, then f must change sign or must be identically zero.
In the present paper we focus on the case χ(M ) < 0, so M is a surface of genus greater than one and K < 0. The complementary cases χ(M ) ≥ 0-i.e., the cases where M = S 2 or M = T , the 2-torus-will be discussed briefly at the end of this introduction, and we also refer the reader to [19,18,2,8] and the references therein.Multiplying equation (1.2) with the factor e −2u and integrating over M with respect to the measure µ ḡ , we get the following necessary condition-already mentioned by Kazdan and Warner in [11]-for the average f := 1 volḡ M f (x)dµ ḡ (x), with vol ḡ := M dµ ḡ (x): x) dµ ḡ (x) < 0. (1.4) This condition is not sufficient.Indeed, it has already been pointed out in [11,Theorem 10.5] that in the case χ(M ) < 0 there always exist functions f ∈ C ∞ (M ) with f < 0 and the property that (1.2) has no solution.
We recall that solutions of (1.2) can be characterised as critical points of the functional x) dµ ḡ (x). (1.5) Under the assumption χ(M ) < 0, i.e., K < 0, the functional E f is strictly convex and coercive on H 1 (M, ḡ) if f ≤ 0 and f does not vanish identically.Hence, as noted in [7], the functional E f admits a unique critical point u f ∈ H 1 (M, ḡ) in this case, which is a strict absolute minimiser of E f and a (weak) solution of (1.2).
The situation is more delicate in the case where f λ = f 0 + λ, where f 0 ≤ 0 is a smooth, nonconstant function on M with max x∈M f 0 (x) = 0, and λ > 0. In the case where λ > 0 sufficiently small (depending on f 0 ), it was shown in [7] and [1] that the corresponding functional E f λ admits a local minimiser u λ and a further critical point u λ ̸ = u λ of mountain pass type.These results motivate our present work, where we suggest a new flow approach to the prescribed Gausss curvature problem in the case χ(M ) < 0. It is important to note here that there is an intrinsic motivation to formulate the static problem in a flow context.Typically, elliptic theories are regarded as the static case of the corresponding parabolic problem; in that sense, many times the better-understood elliptic theory has been a source of intuition to generalise the corresponding results in the parabolic case.Examples of this feedback are minimal surfaces/mean curvature flow, harmonic maps/solutions of the heat equation, and the Uniformisation Theorem/the two-dimensional normalised Ricci flow.
In this spirit, a flow approach to (1.2), the so-called prescribed Gauss curvature flow, was first introduced by Struwe in [19] (and [2]) for the case M = S 2 with the standard background metric and a positive function f ∈ C 2 (M ).More precisely, he considers a family of metrics (g(t, •)) t≥0 which fulfils the initial value problem ∂ t g(t, x) = 2(α(t)f (x) − K g(t,•) (x))g(t, x) in (0, T ) × M ; (1.6) with . (1.8) This choice of α(t) ensures that the volume of (M, g(t, •)) remains constant throughout the deformation, i.e., M dµ g(t,•) (x) = M e 2u(t,x) dµ ḡ (x) ≡ vol g0 for all t ≥ 0, where g 0 denotes the initial metric on M .Equivalently one may consider the evolution equation for the associated conformal factor u given by g(t, x) = e 2u(t,x) ḡ(x): (1.9) u(0, x) = u 0 (x) on {0} × M. (1.10) Here the initial value u 0 is given by g 0 (x) = e 2u0(x) ḡ(x).The flow associated to this parabolic equation is usually called the prescribed Gauss curvature flow.With the help of this flow, Struwe [19] provided a new proof of a result by Chang and Yang [6] on sufficient criteria for a function f to be the Gauss curvature of a metric g(x) = e 2u(x) g S 2 (x) on S 2 .He also proved the sharpness of these criteria.
In the case of surfaces with genus greater than one, i.e., with negative Euler characteristic, the prescribed Gauss curvature flow was used by Ho in [9] to prove that any smooth, strictly negative function on a surface with negative Euler characteristic can be realised as the Gaussian curvature of some metric.More precisely, assuming that χ(M ) < 0 and that f ∈ C ∞ (M ) is a strictly negative function, he proves that equation (1.9) has a solution which is defined for all times and converges to a metric g ∞ with Gaussian curvature K g∞ satisfying for some constant α ∞ .
While the prescribed Gauss curvature flow is a higly useful tool in the cases where f is of fixed sign, it cannot be used in the case where f is sign-changing.Indeed, in this case we may have M f (x)dµ g(t,•) (x) = 0 along the flow and then the normalising factor α(t) is not well-defined by (1.8).As a consequence, a long-time solution of (1.9) might not exist.In particular, the static existence results of [7] and [1] can not be recovered and reinterpreted with the standard prescribed Gauss curvature flow.
In this paper we develop a new flow approach to (1.2) in the case χ(M ) < 0 for general f ∈ C ∞ (M ), which sheds new light on the results in [7], [1] and [9].The main idea is to replace the multiplicative normalisation in (1.9) by an additive normalisation, as will be described in details in the next chapter.
At this point, it should be noted that the normalisation factor α(t) in the prescribed Gauss curvature flow given by (1.8) is also not the appropriate choice in the case of the torus, where, as noted before, f has to change sign or be identically zero in order to arise as the Gauss curvature of a conformal metric.The case of the torus was considered by Struwe in [18], where, in particular, he used to a flow approach to reprove and partially improve a result by Galimberti [8] on the static problem.In this approach, the normalisation in (1.8) is replaced by .11)With this choice, Struwe shows that for any smooth there exists a unique, global smooth solution u of (1.9) satisfying u(t, In principle, the normalisation (1.11) could also be considered in the case χ(M ) < 0, but then the flow is not volume-preserving anymore, which results in a failure of uniform estimates for solutions of (1.9).Consequently, we were not able to make use of the associated flow in this case.
The paper is organised as follows.In Section 2 we set up the framework for the new variant of the prescribed Gauss curvature flow with additive normalisation, and we collect basic properties of it.In Section 3, we then present our main result on the long-time existence and convergence of the flow (for suitable times t k → ∞) to solutions of the corresponding static problem.In particular, our results show how sign changing functions of the form f λ = f 0 + λ arise depending on various assumptions on the shape of f 0 and on the fixed volume A of M with respect to the metric g(t).Before proving our results on the time-dependent problem, we first derive, in Section 4, some results on the static problem with volume constraint.Most of these results will then be used in Section 5, where the parabolic problem is studied in detail and the main results of the paper are proved.In the appendix, we provide some regularity estimates and a variant of a maximum principle for a class of linear evolution problems with Hölder continuous coefficients.

A New Flow Approach and Some of its Properties
Before introducing the additively rescaled prescribed Gauss curvature flow, we recall an important and highly useful estimate.The following lemma (see e.g.[5,Corollary 1.7]) is a consequence of the Trudinger's inequality [20] which was improved by Moser in [15] (for more details see e.g.[18, Theorem 2.1 and Theorem 2.2]): Lemma 2.1.For a two-dimensional, closed Riemannian manifold (M, ḡ) there are constants η > 0 and for all u ∈ H 1 (M, ḡ) where in view of our assumption that vol ḡ = 1.
As a consequence of Lemma 2.1, we have for every u ∈ H 1 (M, ḡ) and p > 0. Therefore, for a given A > 0, the set is well defined.We also note that ū ≤ since by Jensen's inequality and our assumption that vol ḡ = 1 we have Next, we let f ∈ C ∞ (M ) be a fixed smooth function.As a consequence of (2.2), the energy functional E f given in (1.5) is then well defined and of class C 1 on H 1 (M, ḡ).Moreover, we have We now consider the additively rescaled prescribed Gauss curvature flow given by the evolution equation where α(t) is chosen such that the volume vol g(t) of M with respect to the metric g(t) = e 2u(t) ḡ remains constant along the flow.The latter condition requires that 1 2 vanishes for t > 0 and therefore suggest the definition of α(t) given in (2.11) below.We first note the following observations.
To show 3., we note that replacing f by f + c in (2.9) gives so the equation remains unchanged.

Main Results
In the following, we put The following is our first main result.
and u 0 ∈ C p,A for a given A > 0.
Then the initial value problem (2.9), (2.10) admits a unique global solution Moreover, u is uniformly bounded in the sense that Furthermore, if (t l ) l ⊂ (0, ∞) is a sequence with t l → ∞ as l → ∞, then, after passing to a subsequence, u(t l ) converges in where In other words, u ∞ induces a metric g ∞ with vol g∞ = A and Gauss curvature K g∞ satisfying Some remarks are in order.
Remark 3.2.It follows in a standard way that, under the assumptions of Theorem 3.1, the ω-limit set 3), which are precisely the critical points of the restriction of the energy functional E f to C A .
In particular, the connectedness implies that, if u ∞ in Theorem 3.1 is an isolated critical point in C A , then ω(u 0 ) = {u ∞ } and therefore we have the full convergence of the flow line In particular, (3.5) holds if u ∞ is a strict local minimum of the restriction of E f to C A .
Remark 3.3.For functions f < 0, the convergence of the flow (1.9) is shown in [9].For the additively rescaled flow (2.9) with initial data (2.10) we get convergence for arbitrary functions f ∈ C ∞ (M ).In general we do not have any information about λ and therefore no information about the sign of f λ in Theorem 3.1.On the other hand, more information can be derived for certain functions f ∈ C ∞ (M ) and certain values of A > 0.
(i) In the case where 2), and therefore this also applies to λ in Theorem 3.1 in this case.
(ii) The following theorems show that f λ in Theorem 3.1 may change sign if , so in this case we get a solution of the static problem (1.2) for sign-changing functions f ∈ C ∞ (M ) by using the additively rescaled prescribed Gauss curvature flow (2.9).Theorem 3.4.Let p > 2. For every A > 0 and c > − K A there exists ε = ε(c, A, K) > 0 with the following property. If In particular, if f has zeros on M , then f λ in (3.4) is sign changing.
Under fairly general assumptions on f , we can prove that λ > 0 if A is sufficiently large and u 0 ∈ C p,A is chosen suitably.
Then there exists κ > 0 with the property that for every A ≥ κ there exists u 0 ∈ C p,A such that the value λ defined in (3.3) is positive.
In fact we have even more information on the associated limit u ∞ in this case, see Corollary 4.7 below.It remains open how large λ can be depending on A and f .The only upper bound we have is since we must have fλ so that f λ fulfills the necessary condition (1.4) provided by Kazdan and Warner in [11].

The static Minimisation Problem with Volume Constraint
To obtain additional information on the limiting function u ∞ and the value λ ∈ R associated to it by (3.3) and (3.4), we need to consider the associated static setting for the prescribed Gauss curvature problem with the additional condition of prescribed volume.In this setting, we wish to find, for given f ∈ C ∞ (M ) and A > 0, critical points of the restriction of the functional E f defined in (1.5) to the set C A defined in (2.3).A critical point u ∈ C A of this restriction is a solution of (3.2) for some λ ∈ R, where, here and in the following, we put again In other words, such a critical point induces, similarly as the limit u ∞ in Theorem 3.1, a metric g u with Gauss curvature arises in this context as a Lagrange multiplier and is a posteriori characterised again by In the study of critical points of the restriction of E f to C A , it is natural to consider the minimisation problem first.For this we set We have the following estimates for m f,A : Proof.Let u 0 (A) ≡ 1 2 log(A), so that M e 2u0(A) dµ ḡ = A. Hence u 0 (A) is the (unique) constant function in C A , and This shows (4.1).To show (4.2), we let ε > 0. Since f ∈ C ∞ (M ) and max f ≥ 0 by assumption, there exists an open set Ω ⊂ M with f ≥ −ε on Ω. Next, let ψ ∈ C ∞ (M ), ψ ≥ 0, be a function supported in Ω and with ∥ψ∥ L ∞ (M,ḡ) = 2. Consequently, the set Ω ′ := {x ∈ M | ψ > 1} is a nonempty open subset of Ω, and therefore µ ḡ (Ω ′ ) > 0.
Next we consider the continuous function and we note that h(0) = M dµ ḡ = 1, and that Hence for every A ≥ 1 there exists Since f ≥ −ε on Ω, we thus deduce that Since ε > 0 was chosen arbitrarily, (4.2) follows.
For every ε > 0 there exists κ 0 > 0 with the following property.If A ≥ κ 0 and u ∈ C A is a solution of for some λ ∈ R with E f (u) < εA 2 , then we have λ < ε.
for n → ∞, and let (u n ) n∈N be a sequence of solutions of which are weakly stable in the sense that , where u 0 is the unique solution of Proof.We only need to show that Indeed, assuming this for the moment, we may complete the argument as follows.Suppose by contradiction that there exists ε > 0 and a subsequence, also denoted by (u n ) n∈N , with the property that By (4.8) and the compactness of the embedding C 2,α (M ) → C 2 (M ), we may then pass to a subsequence, still denoted by (u n ) n∈N , with u n → u * in C 2 (M ) for some u * ∈ C 2 (M ).Passing to the limit in (4.5), we then see that u * is a solution of (4.7), which by uniqueness implies that u * = u 0 .This contradicts (4.9), and thus the claim follows.
The proof of (4.8) follows by similar arguments as in [7, p. 1063 f.].Since the framework is slightly different, we sketch the main steps here for the convenience of the reader.We first note that, by the same argument as in [7, p. 1063 f.], there exists a constant C 0 > 0 with Moreover, we let w n := u n − w, and we note that w n satisfies Multiplying this equation by e 2wn and integrating by parts, we obtain Next we claim that also ∥e wn ∥ L 2 (M,ḡ) remains uniformly bounded.Suppose by contradiction that We then set v n := e wn ∥e wn ∥ L 2 (M,ḡ) , and we note that by (4.14).Consequently, we may pass to a subsequence satisfying v n ⇀ v in H 1 (M, ḡ), where v is a constant function with ∥v∥ L 2 (M,ḡ) = 1.(4.17) However, since ) for all n ∈ N by (4.11) and therefore by (4.15), we conclude that the constant function v must vanish identically, contradicting (4.17).Consequently, ∥e wn ∥ L 2 (M,ḡ) remains uniformly bounded, which by (4.14) implies that e wn remains bounded in H 1 (M, ḡ) and therefore in L p (M, ḡ) for any p < ∞.Since e un ≤ ∥e w ∥ L ∞ (M,ḡ) e wn on M for all n ∈ N, it thus follows that also e un remains bounded in L p (M, ḡ) for any p < ∞.Moreover, by (4.10), the same applies to the sequence u n itself.Therefore, applying successively elliptic L p and Schauder estimates to (4.5), we deduce (4.8), as required.
In the proof of the next proposition, we need the following classical interpolation inequality, see e.g.[4].Lemma 4.4 (Gagliardo-Nirenberg-Ladyžhenskaya inequality).For every r > 2, there exists a constant for every ζ ∈ H 1 (M, ḡ).(i) If λ ≤ 0, then u λ is the unique solution of and a global minimum of E f λ .
(ii) If λ ∈ (0, λ ♯ ], then u λ is the unique weakly stable solution of (4.18) in the sense of (4.6), and it is a local minimum of E f λ .
(iii) The curve of functions λ → u λ is pointwisely strictly increasing on M , and so the volume function is continuous and strictly increasing.
Proof.We already know that, for λ ≤ 0, the energy E f λ admits a strict global minimiser u λ which depends smoothly on λ.Moreover, by [1, Proposition 2.4], the curve λ → u λ can be extended as a C 1 -curve to an interval (−∞, λ ♯ ] for some λ ♯ > 0. We also know from [1, Proposition 2.4] that, for λ ∈ (−∞, λ ♯ ], the solution u λ is strongly stable in the sense that Here we note that the function λ → C λ is continuous since u λ depends continuously on λ with respect to the C 2 -norm.Next we prove that, after making λ ♯ > 0 smaller if necessary, the function u λ is the unique weakly stable solution of (4.18) for λ ∈ (0, λ ♯ ].Arguing by contradiction, we assume that there exists a sequence λ n → 0 + and corresponding weakly stable solutions (u n ) n∈N of with the property that u n ̸ = u λn for every n ∈ N. By Proposition 4.3, we know that Combining this fact with (4.20), we deduce that on M for all n ∈ N, which then implies with Hölder's inequality and Lemma 4.4 that with a constant C > 0 independent on M .This contradicts the fact that v n → 0 in H 1 (M ) as n → ∞.The claim thus follows.
It remains to prove that the curve of functions λ → u λ is pointwisely strictly increasing on M .This is a consequence of the uniqueness of weakly stable solutions stated in (ii) and the fact that, as noted in [7], if u λ0 is a solution for some λ 0 ∈ (−∞, λ ♯ ], it is possible to construct, via the method of sub-and supersolutions, for every λ < λ 0 , a weakly stable solution u λ with u λ < u λ0 everywhere in M . Corollary 4.6.Let f ∈ C ∞ (M ) be nonconstant with max x∈M f (x) = 0, and let λ ♯ > 0 be given as in Proposition 4.5.Then there exists κ 1 > 0 with the following property. If 2 , then 0 < λ < λ ♯ , and u is not a weakly stable solution of (4.23), so u ̸ = u λ .Proof.Let κ 0 > 0 be given as in Lemma 4.2 for ε = λ ♯ > 0.Moreover, let with V defined in (4.19).Next, let u ∈ C A be a solution of (4.23) for some λ ∈ R with E f (u) < λ ♯ A 2 .From Lemma 4.2, we then deduce that 0 < λ < λ ♯ , and by Proposition 4.5 (iii) we have u ̸ = u λ .Since u λ is the unique weakly stable solution of (4.23), it follows that u is not weakly stable.
Corollary 4.7.Let p > 2, f ∈ C ∞ (M ) be nonconstant with max x∈M f (x) = 0, and let λ ♯ > 0 be given as in Proposition 4.5.Then there exists κ > 0 with the property that for every A ≥ κ the set is nonempty, and for every of the initial value problem (2.9), (2.10) converges, as t → ∞ suitably, to a solution u ∞ of the static problem (4.23) for some λ ∈ (0, λ ♯ ) which is not weakly stable and hence no local minimiser of E f λ .

Proof of the Main Results
5.1.Preliminaries.In the following, we consider, for fixed T > 0, the spaces )) and L p t H q x := L p ([0, T ]; H q (M, ḡ)).We stress that, although these spaces depend on T , we prefer to use a T -independent notation.We also note that, since T < ∞ and vol ḡ = 1, we have Lemma 5.1 (Sobolev inequality).There exists a constant C S > 0 such that for every T ≤ 1 and every ρ ∈ L ∞ t H 1 x we have ∥ρ∥ 2 Proof.By Lemma 4.4, applied with r = 4, there exists a constant C GNL = C GNL (4) > 0 with the property that, for all T ≤ 1, Hence the first inequality in (5.1) holds with x for all p ∈ [1, ∞] which shows that the RHS in (5.1) is finite.
Now we can turn to the proofs of the main results.

Short-Time Existence.
Let A > 0 and p > 2 be fixed.We are looking for a short-time solution of (2.9), (2.10) with initial value u 0 ∈ C p,A , where C p,A is defined in (3.1).Using the Gauss equation (1.1) we can rewrite (2.9), (2.10) in the following way: where To find a solution of (5.2), (5.3) on a short time interval, we consider the linear equation and use a fixed point argument in the Banach space For this we first observe that equation (5.4) is strongly parabolic for v ∈ X.Furthermore, since p > 2 and M is compact, we have u 0 ∈ C p,A ⊂ H 2 (M, ḡ), and therefore u 0 ∈ C(M ).
For the fixed point argument we fix u 0 ∈ C p,A and set For fixed T > 0 and v ∈ X, we then get, by Proposition 6.2 in the appendix, a unique solution u v ∈ W 2,1 p ((0, T )× M ) of (5.4) which satisfies (5.5) in the initial trace sense.Here W 2,1 p ((0, T ) × M ) denotes the space of functions u ∈ L p ((0, T ) × M ) which have weak derivatives Du, D 2 u and ∂ t u in L p ((0, T ) × M ), so this space is compactly embedded in C(X) by Lemma 6.1 in the appendix.On X R = {U ∈ X | ∥U ∥ X ≤ R}, we now define the function Φ as follows: for v ∈ X R , let Φ(v) =: u v be the unique solution of (5.4), (5.5).First, we show that Φ : Proof.With Proposition 6.4 (ii) we directly get where by (5.7) and since R = ∥u 0 ∥ L ∞ (M,ḡ) + 1, which shows the claim.
We now use Schauder's fixed point Theorem [17] to show the following proposition.
Step 1: First we recall Schauder's Theorem: If H is a nonempty, convex, and closed subset of a Banach space B and F is a continuous mapping of H into itself such that F (H) is a relatively compact subset of H, then F has a fixed point.In our case, B =X = C([0, T ] × M ), H =X R = {u ∈ X | ∥u∥ X = ∥u∥ CtCx ≤ R}, and F =Φ.So to show the existence of a fixed point of Φ in X R , it remains to show that 1. Φ : X R → X R is continuous and 2. Φ(X R ) ⊂ X R is relatively compact.
First, we show that Φ : X R → X R is continuous.For this, let v ∈ X R , and let (v n ) n ⊂ X R be a sequence with ∥v n − v∥ X → 0.Moreover, let u = Φ(v) and u n = Φ(v n ) for n ∈ N. By Proposition 6.2, we know that for n ∈ N with we have e ±2vn → e ±2v and therefore also d n → d in X, which also implies that d n → d in L p t L p x for all p.Moreover, the difference Since also [u n − u](0) = 0, we have, again by Proposition 6.2, p is embedded in X by Lemma 6.1.Together with 5.2, this shows the continuity of Φ : X R → X R .
Next, we show that Φ(X R ) is relatively compact.For this let (u n ) n∈N ⊂ Φ(X R ) be an arbitrary sequence in Φ(X R ), and let v n ∈ X R with Φ(v n ) = u n for n ∈ N. So, by definition of Φ and by Proposition 6.2, we see that for n ∈ N. Hence (u n ) n∈N is uniformly bounded in W 2,1 p ((0, T ) × M ).Using now that W 2,1 p ((0, T ) × M ) is compactly embedded in X by Lemma 6.1, we conclude the claim.We have thus proved that Φ has a fixed point u in X R , which then is a (strong) solution u ∈ W 2,1 p ((0, T ) × M ) of (5.2), (5.3).
We now assume by contradiction that u is not continuous at t = 0 with respect to the H 1 (M, ḡ)-norm.Then there exists a sequence (t n ) n∈N in (0, T ) and ε > 0 with t n → 0 + as n → ∞ and for all n ∈ N. (5.9) Since ∥u(t n )∥ 2 H 1 (M,ḡ) = ϕ(t n ) remains bounded as n → ∞, we conclude that, passing to a subsequence, the sequence u(t n ) converges weakly in H 1 (M, ḡ) and therefore strongly in L 2 (M, ḡ).Since the strong L 2 -limit of u(t n ) must be u 0 = u(0) as a consequence of the fact that u ∈ X, we deduce that u(t n ) ⇀ u 0 weakly in H 1 (M, ḡ) as n → ∞.Combining this information with Proposition 6.2 from the appendix, we deduce that lim sup (5.10) and therefore ∥u(t n )∥ H 1 (M,ḡ) → ∥u 0 ∥ H 1 (M,ḡ) .Note here that this part of Proposition 6.2 applies since u solves (5.4), (5.5) for some α > 0. From (5.10) and the uniform convexity of the Hilbert space H 1 (M, ḡ), we conclude that u(t n ) → u 0 strongly in H 1 (M, ḡ), contrary to (5.9).
Proof.Let u 1 , u 2 ∈ X ∩ C ∞ ((0, T ) × M ) be two solutions of (5.2), (5.3).The difference u )dµ ḡ for t ∈ (0, T ). (5.11) In the following, the letter C denotes different positive constants.Multiplying (5.11) with 2u and integrating over M gives with functions V ∈ L p ((0, T ) × M ) ∩ C ∞ ((0, T ) × M ) and ρ ∈ L ∞ (0, T ).Here we used the Sobolev embeddings ).Multiplying (5.11) with −2∆u and integrating over M yields where we used first Hölder's inequality with α = 2p p−2 , then Young's inequality and finally Sobolev embeddings again.Here we note that, by making C > 0 larger if necessary, we may assume that the constants are the same in (5.13) and (5.14).Combining these estimates gives for t ∈ (0, T ) ( with the function g ∈ L 1 (0, T ) given by g Integrating and using the fact that u ∈ C([0, T ), H 1 (M, ḡ)) by Proposition 5.3 with u(0) = u 1 (0) − u 2 (0) = 0, we see that In this section, we wish to show that the (unique) local solution of the initial value problem (5.2), (5.3) for small T > 0 can be extended to a global und uniformly bounded solution defined for all positive times.We first need the following local boundedness property on open time intervals.
Proof.Since K < 0, we have 2), where by (2.12).Hence the function v = −u satisfies For fixed k ∈ N the continuous function e 2v is then bounded from below by a positive constant on the compact set [0, T k ] × M .Therefore Proposition 6.4 (ii) from the appendix implies that Letting k → ∞, we deduce that (5.17) In order to derive an upper bound for u, we now observe that on M for t ∈ [0, T ).Applying Proposition 6.4 (ii) in the same way as above therefore gives Combining (5.17) and (5.18) yields sup as claimed in (5.18).
In the next lemma, with the help of (2.17), we turn (5.16) into a uniform estimate for all time.
Lemma 5.7.Let u be the global, smooth solution of the initial value problem (5.2), (5.3).Then we have ) > 0 which is increasing in its second variable.
Proof.We argue similarly as in the proof of [18,Lemma 2.5].By using the fact that u(t) is a volume preserving solution of (5.2) with u(0) = u 0 ∈ C p,A and therefore M e 2u(t) dµ ḡ ≡ A, we get with (2.4) and the fact that K < 0 that (5.20) For the function we then obtain, by combining (5.20) with (2.17), the estimate Hence, for any T > 0 we find t T ∈ [T, T + 1] such that with constants Here we used (2.6).So, at time t T we get with (2.7), Hölders inequality, Young's inequality, (2.15), and (5.23) that (5.24) with constants d i = d i (u 0 ), i ∈ {3, 4}.Here the constants ν i (u 0 , 3), i ∈ {0, 1} are given in (2.15).Furthermore, with Sobolev's embedding theorem we have W 2, 3 2 (M ) ⊂ C 0, 2 3 ⊂ L ∞ (M, ḡ).Therefore we get with Poincaré's inequality, the Calderón-Zygmund inequality for closed surfaces, and with (5.24) that (5.25) with constants d i > 0, i ∈ {5, 6, 7} and (2.14) we therefore obtain the uniform bound Upon shifting time by t T , we therefore get from Lemma 5.5 sup (5.27) Since M is increasing in its first and second variables by Lemma 5.5, we see that N is increasing in ∥f ∥ L ∞ (M,ḡ) , as claimed.Since T > 0 was arbitrary, the claim follows.
5.5.Convergence of the Flow.Let f ∈ C ∞ (M ), A > 0, p > 2 and u 0 ∈ C p,A as before, and let u denote the global, smooth solution of the initial value problem (5.2), (5.3).In this section we shall show that for a suitable sequence t l → ∞, l → ∞, the associated sequence of metrics g(t l ) tends to a limit metric g ∞ = e 2u∞ ḡ with Gauss curvature K g∞ , which then implies that K g∞ = f − α ∞ with a constant α ∞ .Afterwards, we shall have a closer look at this constant α ∞ .By (5.22), we know that, for a suitable sequence t l → ∞, l → ∞ we have (5.28) We can strengthen this observation as follows.
With Lemma 4.4, applied with r = 3, C GNL = C GNL (3) > 0, (5.2) and Lemma 5.7 we can furthermore estimate where we used Young's inequality and the fact that Combining (5.32) and (5.33) and using that G(t) ≥ 0 gives ( which shows the claim. To prove now the convergence of the flow, we first note u(t) is uniformly (in t ∈ (0, ∞)) bounded in H 1 (M, ḡ) by Proposition 2.25.and Lemma 5.8.We now consider a sequence t l → ∞, l → ∞ and the associated sequence of functions u l := u(t l ).This sequence is bounded in H 1 (M, ḡ), hence there exists a subsequence, again denoted by (u l ) l , with u l → u ∞ weakly in H 1 (M, ḡ) and therefore strongly in L 2 (M, ḡ).Furthermore with (2.12) we know that α l := α(t l ) → α ∞ as l → ∞ after passing again to a subsequence.Moreover we claim that e ±u l → e ±u∞ (as l → ∞) in L p (M, ḡ) for any 2 ≤ p < ∞.Indeed, using Lemma 5. we find that Replacing u l by −u l we get also e −u l → e −u∞ in L p (M, ḡ) as l → ∞ for any p < ∞.Furthermore, we have Since moreover e 2u l ∂ t u l → 0 in L 2 (M, ḡ) as l → ∞ with Lemma 5.7 and Lemma 5.8, the evolution equation (5.2) yields for some p > 2, and let u 0 ∈ W 2,p (M, ḡ).
Then the initial value problem (6.1), (6.2) has a unique strong solution u ∈ W 2,1 p (Ω T ).Moreover, u satisfies the estimate ∥u∥ with a constant C > 0 depending only on ∥a∥ L ∞ (Ω T ) , ∥c∥ L ∞ (Ω T ) and a T .Moreover, C does not increase after making T smaller.If, moreover, a, c, d ∈ C α (Ω T ) for some α > 0, then u ∈ C(Ω T ) ∩ C 2,1 (Ω T ) is a classical solution of (6.1), (6.2), and we have the inequality Proof.In the following, the letter C stands for various positive constants depending only on ∥a∥ L ∞ (Ω T ) , ∥c∥ L ∞ (Ω T ) , and a T , and which do not increase after making T smaller.
Consequently, combining (6.6) and (6.7), and using an interpolation estimate for Du, we find that as claimed in (6.3).
Step 3: In the general case, we consider a sequence a n ∈ C 0,1 (Ω T ) with a n → a in C(Ω T ), and we let u n denote the associated solutions of (6.1), (6.2) with a replaced by a n .As in the end of the proof of Proposition 6.2, we then find that, after passing to a subsequence, u n ⇀ ũ in W 2,1 p (Ω T ), where ũ is a solution of (6.1), (6.2).By uniqueness, we have u = ũ.Moreover, since u n ≤ 0 for all n by Step 3, we have u = ũ ≤ 0, as required.(ii) We consider the function v ∈ W 2,1 p (Ω T ) given by v(t, x) = u(t, x)−∥u + 0 ∥ L ∞ (M,ḡ) −td T , which, by assumption, satisfies (6.1), (6.2) with c ≡ 0, d − d T in place of d and u 0 − ∥u + 0 ∥ L ∞ (M,ḡ) in place of u 0 .Then (i) yields v ≤ 0 in Ω T , and therefore u satisfies (6.13).

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