A gradient flow for the prescribed Gaussian curvature problem on a closed Riemann surface with conical singularity
aa r X i v : . [ m a t h . A P ] J un A gradient flow for the prescribed Gaussian curvature problem ona closed Riemann surface with conical singularity
Yunyan Yang
Department of Mathematics, Renmin University of China, Beijing 100872, P. R. China
Abstract
In this note, we prove that the abstract gradient flow introduced by Baird-Fardoun-Regbaoui [2]is well-posed on a closed Riemann surface with conical singularity. Long time existence andconvergence of the flow are proved under certain assumptions. As an application, the prescribedGaussian curvature problem is solved when the singular Euler characteristic of the conical sur-face is non-positive.
Keywords: prescribed Gaussian curvature problem, conical singularity, gradient flow
1. Introduction
Let Σ be a closed Riemann surface, g be a smooth metric and κ be its Gaussian curvature. If˜ g = e u g for some smooth function u , then the Gaussian curvature of ˜ g satisfies ˜ κ = e − u ( ∆ g u + κ ),where ∆ g is the Laplace-Beltrami operator. For a given function K : Σ → R , can one find a metric˜ g = e u g having K as its Gaussian curvature? This problem is equivalent to the solvability of theequation ∆ g u + κ − Ke u = . (1)Integration by parts and the Gauss-Bonnet formula imply that necessarily K must have the samesign as the topological Euler characteristic χ ( Σ ) somewhere and in the case χ ( Σ ) =
0, either K isidentically zero or changes sign. It is natural to ask if this condition is also su ffi cient to guaranteea solution.In the case χ ( Σ ) <
0, via the method of upper and lower solutions, it was shown by Kazdan-Warner [24] that if K ≤ K .
0, then (1) has a solution. Suppose that K ≤ sup Σ K = K .
0, and λ ∈ R . Using a variational method, Ding-Liu [20] proved the following: Replacing K by K + λ in (1), one finds some constant λ ∗ > < λ < λ ∗ , then (1) has at least twodi ff erent solutions; if λ = λ ∗ , then (1) has at least one solution; while if λ > λ ∗ , then (1) has nosolution. In the case χ ( Σ ) =
0, the problem was completely solved. It was proved by Berger [6]that if K ≡ K changes sign and R Σ Ke v dv g <
0, where v is a solution of ∆ g v = − κ , then (1)has a solution. Later Kazdan-Warner [24] pointed out that the above assumptions on K is alsonecessary. If χ ( Σ ) > Σ is either the projective space RP or the 2-sphere S . In the case of RP , Email address: [email protected] (Yunyan Yang)
Preprint submitted to *** November 13, 2018 t was shown by Moser [31] that (1) has a solution provided that sup Σ K > K ( p ) = K ( − p )for all p ∈ S . While the problem on S is much more complicated and known as the Nirenbergproblem. Moser’s result was extended by Chang-Yang [12] to reflected symmetric function K under further assumptions. For rotationally symmetric function K , su ffi cient condition was givenby Chen-Li [15] and Xu-Yang [37]. Concerning more general functions K , we refer the readerto [10, 11, 14].Also various flows have been employed to attack the problem. In [22], The Ricci flow wasintroduced by Hamilton to find a solution of (1), where K is a constant. His result was latercompleted by Chow [19]. The Calabi flow was investigated by Bartz-Struwe-Ye [5] and Struwe[34]. While in [35], Struwe used the Gaussian curvature flow to reprove Chang-Yang’s results[12]. For further developments of this flow, we refer the reader to Brendle [8, 9], Ho [23] andZhang [45]. Assuming that the initial metric g has constant Gaussian curvature κ . Baird-Fardoun-Regbaoui [2] proposed an abstract gradient flow, through which g ( t ) converges to a metric havingthe prescribed Gaussian curvature. This method solved (1) perfectly in the case χ ( Σ ) ≤ χ ( Σ ) > Σ be a closed Riemann surface as before. A metric g is said to be a conformal metric havingconical singularity of order β > − p ∈ Σ , if in a local holomorphic coordinate with z ( p ) = u which is continuous and C away from zero such that g = e u | z | β | dz | . If g has conical singularities of order β i > − p i ∈ Σ , i = , · · · , ℓ , we say that g representsa divisor β = P ℓ i = β i p i . Then the pair ( Σ , β ) is called a conical surface, and the correspondingsingular Euler characteristic is written as χ ( Σ , β ) = χ ( Σ ) + ℓ X i = β i , (2)where χ ( Σ ) is the topological Euler characteristic.If χ ( Σ , β ) is nonpositive, the problem can be solved in the variational framework as the caseof smooth metrics. Precisely, it was shown by Troyanov [36] that if χ ( Σ , β ) <
0, then anysmooth negative function is the Gaussian curvature of a unique conformal metric ˜ g representing β . Recently this result has been improved by Zhu and the author [41] by using the variationalmethod of Ding-Liu [20] and Borer-Galimberti-Struwe [7]. In particular, if we assume χ ( Σ , β ) <
0, the background metric g has the Gaussian curvature κ ≡ −
1, and K is a smooth functionsatisfying sup Σ K = K .
0, then there exists a unique function u ∈ C = C ( Σ \ { p , · · · , p ℓ } ) ∩ C ( Σ ) ∩ W , ( Σ , g )such that the metric e u g has the Gaussian curvature K ; moreover, there exists some constant λ ∗ > < λ < λ ∗ , there exist at least two di ff erent functions u , u ∈ C suchthat e u g and e u g have the same Gaussian curvature K + λ ; when λ = λ ∗ , there exists at leastone function u ∈ C such that e u g has the Gaussian curvature K + λ ∗ ; when λ > λ ∗ , there isno function u ∈ W , ( Σ , g ) such that e u g has the Gaussian curvature K + λ . The problem wascompletely solved by Troyanov [36] in the case χ ( Σ , β ) =
0. Namely, there exists a flat metric g representing β ; moreover, a smooth function K is the Gaussian curvature of a metric ˜ g conformalto g if and only if K changes sign and R Σ Kdv g <
0. If χ ( Σ , β ) >
0, then the problem becomes2ery subtle. There is much interesting work concerning this situation, see for examples Troyanov[36], McOwen [29], Chen-Li [16, 17, 18], Luo-Tian [27], Mondello-Panov [30], Bartolucci [3],Bartolucci-De Marchis-Malchiodi [4], Fang-Lai [21] and a very nice survey of Lai [25].Again the Ricci flow is an elegant way to solve the problem on conical surfaces. Yin [42, 43,44] established a basic theory in this regards, and proved long time existence and convergence ofthe flow when χ ( Σ , β ) ≤
0. The convergence in the case χ ( Σ , β ) > χ ( Σ , β ) ≤
0, we obtain the convergence of the flowunder additional assumptions. For the proof of our results, we follow the lines of Baird-Fardoun-Regbaoui [2]. Here the key point is the following observation: the functionals involved are stillanalytic if the background metric has conical singularity.The remaining part of this note is organized as follows: In Section 2, we construct functionalframework and give main results of this note; In Section 3, we prove the analyticity of functionals J and L , and calculate their gradients; In Section 4, we show the long time existence of thegradient flow; In Section 5, a su ffi cient condition for convergence of the flow will be discussed;In Section 6, we prove that when χ ( Σ , β ) ≤
0, the flow converges to the desired solution of theproblem.
2. Notations and main results
Let Σ be a closed Riemann surface, β = P ℓ i = β i p i be a divisor, β i > − i , and g be aconformal metric representing β . Let κ : Σ \ supp β → R be the Gaussian curvature of g , wheresupp β = { p , · · · , p ℓ } . From now on, we assume κ is a constant. Then the Gauss-Bonnet formula(see for example [36]) reads κ Vol g ( Σ ) = Z Σ κ dv g = πχ ( Σ , β ) , where χ ( Σ , β ) is defined as in (2), and dv g denotes the volume element with respect to the conicalmetric g . Clearly there exists a smooth metric g such that g = ρ g , where ρ > Σ , ρ ∈ C ( Σ \ supp β ), and ρ ∈ L r ( Σ ) for some r >
1. Let W , ( Σ , g ) be thecompletion of C ∞ ( Σ ) under the norm k u k W , ( Σ , g ) = Z Σ ( |∇ g u | + u ) dv g ! / , where ∇ g denotes the gradient operator with respect to the metric g . It was observed by Troyanov[36] that W , ( Σ , g ) = W , ( Σ , g ). In particular, W , ( Σ , g ) is a Hilbert space, which is hereafterdenoted by H , with an inner product h u , w i H = Z Σ ( ∇ g u ∇ g w + uw ) dv g . Σ , g ) and theH¨older inequality, one has W , ( Σ , g ) ֒ → L p ( Σ , g ) , ∀ p > . Let ¯ g = e u g be another conical metric representing β and K : Σ \ supp β → R be the Gaussiancurvature of ¯ g . Then K satisfies point-wisely on Σ \ supp β , K = e − u ( κ + ∆ g u ) , (3)where ∆ g denotes the Laplacce-Beltrami operator with respect to the metric g . Obviously, if u isa distributional solution of the equation ∆ g u + κ − Ke u = , (4)then u satisfies (3).Let us define two functionals J : H → R , L : H → R by J ( u ) = Z Σ |∇ g u | dv g + κ Z Σ udv g , (5) L ( u ) = Z Σ Ke u dv g , (6)and a set of functions by S = n u ∈ H : L ( u ) = κ Vol g ( Σ ) = πχ ( Σ , β ) o . (7)The gradients of J and L , ∇J : H → H and ∇L : H → H are defined by h∇J ( u ) , w i H = d J ( u )( w ) = ddt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = J ( u + tw ) , (8) h∇L ( u ) , w i H = d L ( u )( w ) = ddt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = L ( u + tw ) (9)respectively, where u and w are functions taken from H . Hereafter we assume K .
0. It followsthat ∇L ( u ) , u ∈ S . Thus S is a smooth hypersurface in H . A unit normal on S is N ( u ) = ∇L ( u ) k∇L ( u ) k H for any u ∈ S , where k · k H = h· , ·i H . This allows us to consider the gradient of J with respectto the hypersurface S , which is defined by ∇ S J ( u ) = ∇J ( u ) − h∇J ( u ) , N ( u ) i H N ( u ) . (10)The gradient flow of J with respect to the hypersurface S can be written as ∂ t u = −∇ S J ( u ) u (0) = u ∈ S . (11)If the flow exists for all time and converges at infinity, then the limit function u ∞ gives a distri-butional solution of (4). Our first result is an analog of ([2], Theorem 1), namely4 heorem 1. Let Σ be a closed Riemann surface, β = P ℓ i = β i p i be a divisor with β i > − ,i = , · · · , ℓ , and g be a metric representing β . Let J , L and S be defined by (5), (6) and(7) respectively. Suppose that the Gaussian curvature of g is a constant κ , and that K ∈ C ( Σ ) satisfies the condition R Σ Kdv g < χ ( Σ , β ) < R Σ Kdv g < , sup Σ K > χ ( Σ , β ) = Σ K > χ ( Σ , β ) > . (12) Then for any u ∈ S , there exists a unique global solution u ∈ C ∞ ([0 , ∞ ) , H ) of the gradientflow (11), satisfying u ( t ) ∈ S for all t ≥ . Moreover the energy identity Z t k ∂ s u ( s ) k ds + J ( u ( t )) = J ( u ) . (13) holds for all t > . If χ ( Σ , β ) ≤
0, then we have the convergence of the flow, an analog of ([2], Theorem 2).
Theorem 2.
Let u ∈ S and u : [0 , ∞ ) → H be given as in Theorem 1. In the case χ ( Σ , β ) = ,there exists a u ∞ ∈ W , r ( Σ , g ) ∩ C α ( Σ ) for some r > and < α < such that u ( t ) converges tou ∞ in H as t → ∞ , moreover u ∞ + τ is a distributional solution of (4) for some constant τ ; In thecase χ ( Σ , β ) < , there exists a positive constant ǫ depending only on K − ( x ) = max {− K ( x ) , } and the conical metric g such that if u satisfiese γ k u k H sup x ∈ Σ K ( x ) ≤ ǫ , (14) where γ > is a constant depending only on g, then u ( t ) converges in H to a distributionalsolution u ∞ of (4) as t → ∞ . We remark that if K ( x ) ≤
0, then the hypothesis (14) is obviously satisfied by all u ∈ H .Finally, as an interesting application of Theorem 2, we have the following: Corollary 3.
Suppose K ∈ C ( Σ ) and R Σ Kdv g < . If in addition sup x ∈ Σ K ( x ) > in the case χ ( Σ , β ) = , or sup x ∈ Σ max { K ( x ) , } is su ffi ciently small in the case χ ( Σ , β ) < , then there existsa conformal metric ˜ g representing β and having K as its Gaussian curvature.
3. Preliminaries
In this section, we first show the analyticity of the functionals J and L , and then calculatetheir gradients. Lemma 4.
The functionals J : H → R and L : H → R are analytic.Proof. Let u , h ∈ H be fixed. Clearly J has the following Taylor expansion (see for exampleChang [13], Theorem 1.4 of Chapter 1) J ( u + h ) = n X k = J ( k ) ( u ) h ( k ) k ! + R n ( u , h ) h ( n ) , (15)5here J (0) ( u ) = J ( u ), h ( k ) stands for ( h , · · · , h | {z } k ), k = , , , · · · , and R n ( u , h ) satisfies R n ( u , h ) = Z (1 − t ) n − ( n − n J ( n ) ( u + th ) − J ( n ) ( u ) o dt . (16)One easily computes when n ≥ J ( n ) ( u ) h ( n ) = ∂ n ∂ t · ∂ t n J ( u + t h + · · · + t n h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = ··· = t n = = , J ( n ) ( u + th ) h ( n ) = ∂ n ∂ t · ∂ t n J ( u + th + t h + · · · + t n h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = ··· = t n = = . Hence we have lim n →∞ R n ( u , h ) h ( n ) = . (17)Combining (15) and (17), we conclude that J : H → R is analytic.Similar to (15), we have L ( u + h ) = n X k = L ( k ) ( u ) h ( k ) k ! + R L n ( u , h ) h ( n ) , (18)where R L n ( u , h ) h ( n ) is an analog of (16) with J replaced by L . In view of (6), we have for all n ∈ N , t ∈ [0 , L ( n ) ( u ) h ( n ) = Z Σ Ke u h n dv g , L ( n ) ( u + th ) h ( n ) = Z Σ Ke u + th ) h n dv g . Clearly there holds for all t ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) L ( n ) ( u + th ) − L ( n ) ( u ) (cid:17) h ( n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Σ | K | e | u | + | h | ) | h | n dv g ≤ Z Σ K e | u | + | h | ) dv g ! / Z Σ h n dv g ! / . (19)It follows that |R L n ( u , h ) h ( n ) | ≤ Z Σ K e | u | + | h | ) dv g ! / Z Σ h n dv g ! / n ! = √ n ! Z Σ K e | u | + | h | ) dv g ! / Z Σ h n n ! dv g ! / ≤ √ n ! Z Σ K e | u | + | h | ) dv g ! / Z Σ e h dv g ! / (20)Since u and h are fixed functions in H , by a singular Trudinger-Moser inequality ([36], Theorem6), both e | u | + | h | ) and e u belong to L p ( Σ , g ) for any p >
1. Note also K ∈ C ( Σ ). Then it followsfrom (20) that lim n →∞ R L n ( u , h ) h ( n ) = . L : H → R is analytic. (cid:3) Let I be an identity operator. We now define a map ( ∆ g + I ) − : L ( Σ , g ) → H in thefollowing way. For any f ∈ L ( Σ , g ), we say u = ( ∆ g + I ) − f ∈ H provided that ( ∆ g + I ) u = f .Though in our setting, the metric g has conical singularity, the existence and uniqueness of u follows from the Lax-Milgram theorem. Thus the map ( ∆ g + I ) − is well defined. Moreover( ∆ g + I ) − is a linear map, which follows from the linearity of ∆ g + I . Now we have Lemma 5.
The gradients of J and L at u ∈ H are calculated by ∇J ( u ) = u − ( ∆ g + I ) − ( u − κ ) , (21) ∇L ( u ) = ( ∆ g + I ) − ( Ke u ) . (22) Proof.
On one hand, integration by parts gives h∇J ( u ) , w i H = Z Σ (cid:16) ∇ g ∇J ( u ) ∇ g w + ∇J ( u ) w (cid:17) dv g = Z Σ ( ∆ g + I ) ∇J ( u ) wdv g . (23)On the other hand, d J ( u )( w ) = ddt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = J ( u + tw ) = Z Σ ( ∇ g u ∇ g w + κ w ) dv g = Z Σ ( ∆ g u + κ ) wdv g . (24)Combining (8), (23) and (24), we have( ∆ g + I ) ∇J ( u ) = ∆ g u + κ = ( ∆ g + I ) u − ( u − κ ) , which leads to ( ∆ g + I )( ∇J ( u ) − u ) = − ( u − κ ) . Then (21) follows immediately.To calculate ∇L ( u ), we firstly have an analog of (23), h∇L ( u ) , w i H = Z Σ ( ∆ g + I ) ∇L ( u ) wdv g . Secondly we have d L ( u )( w ) = ddt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = L ( u + tw ) = Z Σ Ke u wdv g . Finally, in view of (9), we obtain (22). (cid:3)
4. Long time existence and energy identity
In this section, we prove Theorem 1 by following the lines of Baird-Fardoun-Regbaoui [2].
Proof of Theorem 1.
By (22), we have ∇L ( u ) , u ∈ H since K .
0. We set F ( u ) = −∇J ( u ) + h∇J ( u ) , ∇L ( u ) i H ∇L ( u ) k∇L ( u ) k H . (25)7y Lemma 4 and the fact ∇L ( u ) , u ∈ H , we conclude that F ∈ C ∞ ( H , H ). Thusfrom the classical Cauchy-Lipschitz theorem ([13], Theorem 1.9 of Chapter 1), there exists some T > u ∈ C ∞ ([0 , T ); H ) is a solution of ∂ t u = F ( u ) u (0) = u ∈ S , (26)or equivalently (11). In view of (25), we have kF ( u ) k H ≤ k∇J ( u ) k H . This together with (21) leads to kF ( u ) k H ≤ C k u k H + C . Here and in the sequel, we often denote various constants by the same C . This together with theequation (26) implies that ∂ t k u k H = h ∂ t u , u i H ≤ C k u k H + C , which leads to ∂ t (cid:16) e − Ct k u ( t ) k H (cid:17) ≤ Ce − Ct . Integrating this inequality from 0 to t < T , one has k u ( t ) k H ≤ (1 + k u k H ) e CT / . (27)It follows from (27) that u can be extended for all t ∈ [0 , ∞ ).By (25) and (26), we calculate ∂ t L ( u ( t )) = h∇L ( u ( t )) , ∂ t u i H = h∇L ( u ( t )) , F ( u ) i H = . Then we have for all t ∈ [0 , ∞ ), L ( u ( t )) ≡ L ( u ) = πχ ( Σ , β )and thus u ( t ) ∈ S . We now prove the energy identity (13). By (10), k ∂ t u k H = −h∇J ( u ) , ∂ t u i H + h∇J ( u ) , N ( u ) i H hN ( u ) , ∂ t u i H . Noting that hN ( u ) , ∂ t u i H = k∇L ( u ) k − H ∂ t L ( u ) = , we have k ∂ t u k H = −h∇J ( u ) , ∂ t u i H = − ∂ t J ( u ) . (28)Integrating (28) from 0 to t , we obtain Z t k ∂ s u ( s ) k H ds = J ( u ) − J ( u ( t )) . This ends the proof of the Theorem. (cid:3) . A su ffi cient condition for convergence In this section, we shall prove that if the solution u ( t ) of (11) is uniformly bounded in H ,then the flow must converge in H . Precisely we have the following: Proposition 6.
Let u : [0 , ∞ ) → H be the solution of (11). Suppose that for all t ∈ [0 , ∞ ) , thereexists a constant C satisfying k u ( t ) k H ≤ C . (29) Then there exists some function u ∞ ∈ W , r ( Σ , g ) ∩ C α ( Σ ) for some r > and < α < , such thatu ( t ) converges to u ∞ in H as t → ∞ . Moreover, if χ ( Σ , β ) , , then u ∞ is a solution of (4); if χ ( Σ , β ) = , then u ∞ + c is a solution of (4) for some constant c.Proof. By (13) and (29), there exists a constant C depending only on C and κ such that Z ∞ k ∂ s u ( s ) k H ds ≤ J ( u ) + C . As a consequence, there is a sequence t j → ∞ satisfying k ∂ t u ( t j ) k H = k∇ S J ( u ( t j )) k H → j → ∞ . Since k u ( t j ) k H ≤ C for all j , there would be some u ∞ ∈ H such that up to asubsequence, u ( t j ) ⇀ u ∞ weakly in H (30) u ( t j ) → u ∞ strongly in L q ( Σ , g ) , ∀ q > . (31)Moreover, the singular Trudinger-Moser inequality ([36], Theorem 6) implies that for any γ > C depending only on γ and the conical metric g such that Z Σ e γ u ( t j ) dv g ≤ C . (32) Claim 1.
There holds u ∞ ∈ S . To see this, we have by the mean value theorem Z Σ K ( e u ( t j ) − e u ∞ ) dv g = Z Σ Ke ξ (2 u ( t j ) − u ∞ ) dv g , where ξ lies between 2 u ( t j ) and 2 u ∞ . Clearly e ξ ≤ e u ( t j ) + e u ∞ . Thus in view of (32), we estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Σ K ( e u ( t j ) − e u ∞ ) dv g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Σ | K | Z Σ ( e u ( t j ) + e u ∞ ) dv g ! / Z Σ ( u ( t j ) − u ∞ ) dv g ! / ≤ C Z Σ ( u ( t j ) − u ∞ ) dv g ! / . This together with (31) and the fact that u j ∈ S leads to Z Σ Ke u ∞ dv g = lim j →∞ Z Σ Ke u ( t j ) dv g = πχ ( Σ , β ) . u ∞ ∈ S and thus Claim 1 follows. Claim 2.
There holds ∇ S J ( u ∞ ) = and u ( t j ) → u ∞ in H as j → ∞ . In view of (10), one has ∇ S J ( u ( t )) = ∇J ( u ( t )) − h∇J ( u ( t )) , ∇L ( u ( t )) i H ∇L ( u ( t )) k∇L ( u ( t )) k H . (33)We first prove that ∇ S J ( u ( t j )) converges to ∇ S J ( u ∞ ) weakly in H as j → ∞ . To see this, itsu ffi ces to prove that as j → ∞ , ∇J ( u ( t j )) ⇀ ∇J ( u ∞ ) weakly in H , (34) ∇L ( u ( t j )) ⇀ ∇L ( u ∞ ) weakly in H , (35) h∇J ( u ( t j )) , ∇L ( u ( t j )) i H → h∇J ( u ∞ ) , ∇L ( u ∞ ) i H , (36) k∇L ( u ( t j )) k H → k∇L ( u ∞ ) k H . (37)In view of (21), we have ∇J ( u ( t )) = u ( t ) − ( ∆ g + I ) − ( u ( t ) − κ ) . (38)For any φ ∈ H , one calculates h ( ∆ g + I ) − ( u ( t j ) + κ ) , φ i H = Z Σ ∇ g (cid:16) ( ∆ g + I ) − ( u ( t j ) + κ ) (cid:17) ∇ g φ dv g + Z Σ ( ∆ g + I ) − ( u ( t j ) + κ ) φ dv g = Z Σ ( ∆ g + I ) (cid:16) ( ∆ g + I ) − ( u ( t j ) + κ ) (cid:17) φ dv g = Z Σ ( u ( t j ) + κ ) φ dv g . This together with (30), (31) and (38) leads to (34).In view of (22), ∇L ( u ( t )) = ( ∆ g + I ) − ( Ke u ( t ) ) . (39)For any φ ∈ H , one has as j → ∞ , h ( ∆ g + I ) − ( Ke u ( t j ) ) , φ i H = Z Σ Ke u ( t j ) φ dv g → Z Σ Ke u ∞ φ dv g = h ( ∆ g + I ) − ( Ke u ∞ ) , φ i H . This together with (39) leads to (35).Let f j = ( ∆ g + I ) − ( Ke u ( t j ) ), or equivalently ( ∆ g + I ) f j = Ke u ( t j ) . Then standard ellipticestimates lead to that f j is bounded in W , r ( Σ , g ) for some r > H .Up to a subsequence one may assume ( ∆ g + I ) − ( Ke u ( t j ) ) converges to ( ∆ g + I ) − ( Ke u ∞ ) in H .10imilarly as before, one calculates h∇J ( u ( t j )) , ∇L ( u ( t j )) = Z Σ ∇ g ( ∆ g + I ) − u ( t j ) ∇ g ( ∆ g + I ) − ( Ke u ( t j ) ) dv g + Z Σ ( ∆ g + I ) − u ( t j )( ∆ g + I ) − ( Ke u ( t j ) ) dv g = Z Σ u ( t j )( ∆ g + I ) − (cid:16) Ke u ( t j ) (cid:17) dv g → Z Σ u ∞ ( ∆ g + I ) − ( Ke u ∞ ) dv g = h∇J ( u ∞ ) , ∇L ( u ∞ ) i H . This is exactly (36). As for (37), one has a strong estimate k∇L ( u ( t j )) k H = Z Σ Ke u ( t j ) ( ∆ g + I ) − ( Ke u ( t j ) ) dv g → Z Σ Ke u ∞ ( ∆ g + I ) − ( Ke u ∞ ) dv g = k∇L ( u ∞ ) k H . (40)Therefore we have proved (34)-(37), and thus ∇ S J ( u ( t j )) converges to ∇ S J ( u ∞ ) weakly in H .As a consequence k∇ S J ( u ∞ ) k H = lim j →∞ h∇ S J ( u ( t j )) , ∇ S J ( u ∞ ) i H ≤ lim j →∞ k∇ S J ( u ( t j )) k H k∇ S J ( u ∞ ) k H = . This immediately leads to ∇ S J ( u ( t j )) converges in H to ∇ S J ( u ∞ ) =
0. It follows from (40)that ∇L ( u ( t j )) converges in H to ∇L ( u ∞ ). Therefore, in view of (33) and (38), we obtain u j converges in H to u ∞ . This concludes Claim 2.By (33), (38) and (39), the equation ∇ S J ( u ∞ ) = ∆ g u ∞ + κ = c ∞ Ke u ∞ (41)for some constant c ∞ . By elliptic estimates, we conclude that u ∞ ∈ W , r ( Σ , g ) ∩ C α ( Σ ) for some r > < α <
1. If χ ( Σ , β ) ,
0, then we have by integrating (41), the Gauss-Bonnet formulaand Claim 1 2 πχ ( Σ , β ) = Z Σ κ dv g = c ∞ Z Σ Ke u ∞ dv g = πχ ( Σ , β ) c ∞ . It follows that c ∞ = u ∞ is a distributional solution of (4). If χ ( Σ , β ) =
0, then κ = e − u ∞ , we have − Z Σ e − u ∞ |∇ g u ∞ | dv g = c ∞ Z Σ Kdv g , which together with (12) implies that c ∞ >
0. Then u ∞ + log c ∞ is a distributional solution of (4).Repeating the same argument of ([2], Pages 25-27), one can derive a Lojasiewicz-Simoninequality and then use it to obtain lim t →∞ k u ( t ) − u ∞ k H = . This completes the proof of the proposition. (cid:3) . Convergence of the flow In this section, we prove Theorem 2 by using Proposition 6. The key point is to prove that k u ( t ) k H ≤ C for all t ∈ [0 , ∞ ) under appropriate conditions. Suppose χ ( Σ , β ) =
0. Since κ is a constant, it followsfrom the Gauss-Bonnet formula that κ =
0. In view of (21), on calculates ∆ g u ( t ) = ( ∆ g + I ) ∇J ( u ( t )) . Integration by parts gives Z Σ ∇J ( u ( t )) dv g = , which leads to h∇J ( u ( t )) , i H = . (42)In view of (22), we have Ke u ( t ) = ( ∆ g + I ) ∇L ( u ( t )) . Since u ( t ) ∈ S , we have by integrating by parts Z Σ ∇L ( u ( t )) dv g = Z Σ Ke u ( t ) dv g = . Hence h∇L ( u ( t )) , i H = . (43)It follows from (42) and (43) that ∂ t Z Σ u ( t ) dv g = Z Σ ∂ t udv g = h ∂ t u , i H = . Then there exists a constant C such that Z Σ u ( t ) dv g ≡ C . Using the Poincare inequality, we obtain Z Σ u dv g ≤ C Z Σ |∇ g u | dv g + C . (44)By (13), there holds J ( u ( t )) ≤ J ( u ), or equivalently Z Σ |∇ g u | dv g ≤ Z Σ |∇ g u | dv g . (45)Combining (44) and (45), we obtain k u ( t ) k H ≤ C for some constant C . This together with Proposition 6 completes the proof of the theorem in thecase χ ( Σ , g ) =
0. 12 .2. The negative case
We first have a Poincar´e inequality on conical surfaces.
Lemma 7.
For all u ∈ H , there holds Z Σ u dv g ≤ λ g ( Σ ) Z Σ |∇ g u | dv g + g ( Σ ) Z Σ udv g ! , where λ g ( Σ ) = inf u ∈ H , R Σ udv g = , u . R Σ |∇ g u | dv g R Σ u dv g . (46) Proof.
Applying a direct method of variation to (46), one finds a function u ∈ H satisfying R Σ u dv g = λ g ( Σ ) = Z Σ |∇ g u | dv g > . Denote u = g ( Σ ) Z Σ udv g . By the definition of λ g ( Σ ), we have for all u ∈ H , Z Σ | u − u | dv g ≤ λ g ( Σ ) Z Σ |∇ g u | dv g . Noting that Z Σ u ( u − u ) dv g = u Z Σ ( u − u ) dv g = , we obtain Z Σ u dv g = Z Σ (cid:16) ( u − u ) + u + u ( u − u ) (cid:17) dv g = Z Σ ( u − u ) dv g + u Vol g ( Σ ) ≤ λ g ( Σ ) Z Σ |∇ g u | dv g + g ( Σ ) Z Σ udv g ! . This gives the desired result. (cid:3)
Next we have the following singular Trudinger-Moser inequality.
Lemma 8.
There exist two constants C and β depending only on ( Σ , g ) such that for all u ∈ H , Z Σ e u dv g ≤ C exp β Z Σ |∇ g u | dv g + g ( Σ ) Z Σ udv g ! . (47) Proof.
Note that g is a conical metric. The inequality (47) follows from that of Troyanov ([36],Theorem 6) (see also Zhu [46] for a critical version). (cid:3)
13e remark that (47) is a weak version of Trudinger-Moser inequality. For related strong ver-sions, we refer the reader to recent works [1, 26, 38, 39, 40] and the references therein.
Proof of Theorem 2 in the negative case.
Having Lemmas 7 and 8 in hand, we can prove an ana-log of ([2], Lemma 2) by using the same method, and then repeating the argument of the proofof ([2], Part ( ii ) of Theorem 2), we conclude the theorem in the case χ ( Σ , β ) < (cid:3) Acknowledgements . This work is supported by National Science Foundation of China(Grant Nos. 11171347, 11471014).
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