A flow approach to the generalized Loewner-Nirenberg problem of the σ_k-Ricci equation
aa r X i v : . [ m a t h . A P ] J a n A FLOW APPROACH TO THE GENERALIZED LOEWNER-NIRENBERGPROBLEM OF THE σ k -RICCI EQUATION A bstract . We introduce a flow approach to the generalized Loewner-Nirenberg problem (1 . − (1 .
7) of the σ k -Ricci equation on a compact manifold ( M n , g ) with boundary. We prove thatfor initial data u ∈ C ,α ( M ) which is a subsolution to the σ k -Ricci equation (1 . . − (3 .
3) has a unique solution u which converges in C loc ( M ◦ ) to the solution u ∞ of the problem (1 . − (1 . t → ∞ .
1. I ntroduction
Let ( M n , g ) be a compact Riemannian manifold with boundary of dimension n ≥
3. Denote M ◦ to be the interior of M . In [10], we considered the Cauchy-Dirichlet problem of the Yamabeflow which starts from a positive subsolution of the Yamabe equation (1 .
1) and converges in C loc ( M ◦ ) to the solution to the Loewner-Nirenberg problem4( n − n − ∆ u − R g u − n ( n − u n + n − = , in M ◦ , (1.1) u ( p ) → ∞ , as p → ∂ M , (1.2)which is originally studied by Loewner and Nirenberg [15] on Euclidean domains, and later byAviles and McOwen [2][3] on general compact manifolds with boundary. A signature feature ofour flow is that it preserves the solution u ( · , t ) as a sub-solution to the Yamabe equation for t > Definition 1.1.
For ( λ , ..., λ n ) ∈ R n and k = , .., n, we define the elementary symmetric func-tions as σ k ( λ , ..., λ n ) = X i <...< i k λ i ... λ i k , and define the cone Γ + k = { Λ = ( λ , ..., λ n ) ∈ R n (cid:12)(cid:12)(cid:12) σ j ( Λ ) > , ∀ j ≤ k } , which is the connected component of the set { σ k > } containing the positive definite cone on R n . We also define Γ − k = − Γ + k . For a symmetric n × n matrix A, σ k ( A ) is defined to be σ k ( Λ ) with Λ = ( λ , ..., λ n ) the eigenvalues of A. The σ k -scalar curvature equation is introduced in [18]. Let ( M n , g ) be a smooth compactRimeannian manifold with boundary of dimension n ≥
3. Denote
Ric g as the Ricci tensor of g . Mathematics Subject Classification.
Primary 53C44; Secondary 35K55, 35R01, 53C21. † Research partially supported by the National Natural Science Foundation of China No. 11701326 and theYoung Scholars Program of Shandong University 2018WLJH85. n [7], for any k = , ..., n , the authors studied the Dirichlet boundary value problem of the σ k equation of − Ric g , in seek of a conformal metric ¯ g = e u g such that Ric ¯ g ∈ Γ − k and σ k ( − Ric ¯ g ) ≡ σ k ( − ¯ g − Ric ¯ g ) = ¯ β k , n in M , (1.3) u = ∂ M , (1.4)where ¯ β k , n = ( n − k nk ! , or equivalently, we have the σ k -Ricci equation σ k ( ¯ ∇ u ) = ¯ β k , n e ku (1.5)where ¯ ∇ u = − Ric g + ( n − ∇ u + ∆ u g + ( n − | du | g − du ⊗ du ) . (1.6)A more interesting result in [7] is that they generalized the Loewner-Nirenberg problem to the σ k -Ricci equation (1 .
5) (see also [5]). They proved that there exists a unique solution u k to (1 . u k ( p ) → + ∞ (1.7)uniformly as p → ∂ M ; moreover, lim p ∈ ∂ M [ u k ( p ) + log( r ( p ))] = p → ∂ M , where r ( p ) is the distance of p to ∂ M . Notice that in [5] Guan gave an alternativeapproach to similar results, using metrics of negative Ricci curvature in the conformal classconstructed in [16] as the background metric. In comparison, the argument in [7] uses a generalbackground conformal metric and concludes the existence of a prescribed σ n -Ricci curvaturemetric of negative Ricci curvature. In this paper, we give a flow approach to the generalizedLoewner-Nirenberg problem to the σ k -Ricci equation (1 .
5) starting from a sub-solution to (1 . . − (3 .
3) of the σ k -Ricci curvature flow.In order to get the lower bound control of the blowing up ratio near the boundary, we need toassume that the boundary data φ could not go to infinity too slowly as t → ∞ . Definition 1.2.
We call a function ξ ( t ) ∈ C ([0 , ∞ )) a low-speed increasing function if, ξ ( t ) > for t ≥ , lim t →∞ ξ ( t ) = ∞ , and there exist two constants T > and τ > such that for t ≥ T , ξ ′ ( t ) ≤ τ. (1.9)Here are some examples of low-speed increasing functions: t α for some 0 < α <
1, log( t ),and finitely many composition of log functions: log ◦ log ◦ ... ◦ log( t ) for t > Theorem 1.3.
Assume ( M n , g ) ( n ≥ is a compact manifold with boundary of C ,α , and ( M , g ) is either a compact domain in R n or with Ricci curvature Ric g ≤ − δ g for some δ ≥ ( n − .Assume u ∈ C ,α ( M ) is a subsolution to (1 . satisfying (3 . at the points x ∈ ∂ M wherev ( x ) = for the function v defined in (3 . . Also, assume φ ∈ C + α, + α ( ∂ M × [0 , T ]) for allT > , φ t ( x , t ) ≥ on ∂ M × [0 , + ∞ ) and φ satisfies the compatible condition (3 . with u .Moreover, assume that there exist a low-speed increasing function ξ ( t ) satisfying (1 . for someT > and τ > , and a constant T > T such that φ ( x , t ) ≥ log( ξ ( t )) for ( x , t ) ∈ ∂ M × [ T , ∞ ) .Then there exists a unique solution u ∈ C , ( M × [0 , + ∞ )) to the Cauchy-Dirichlet problem . − (3 . such that u ∈ C + α, + α ( M × [0 , T ]) for all T > . Moreover, the solution u convergesto a solution u ∞ to the equation (1 . locally uniformly on M ◦ in C , and lim x → ∂ M ( u ∞ ( x ) + log( r ( x ))) = , where r ( x ) is the distance of x to ∂ M. Notice that our assumption on the boundary data φ and the speed that φ → ∞ as t → ∞ is pretty general. When u is a solution to (1 .
5) in a neighborhood of ∂ M , then (3 .
6) holdsautomatically; while the condition (3 .
6) disappears when u is a strict sub-solution to (1 .
5) in aneighborhood of ∂ M . For instance, for any sub-solution u to (1 . u − C is a strict sub-solutionfor any constant C >
0. For the long time existence of the flow, one needs to establish the globala priori estimates on the solution u up to C -norm: both the boundary estimates and the interiorestimates, starting from the L ∞ control by the maximum principle and heavily depending on themonotonicity of u and the control of u t . In particular, u t ≥ u ( · , t ) is a sub-solutionto (1 .
5) for any t ≥
0, which together with the uniform interior upper bound control makes theconvergence possible and gives a natural lower bound of u . For the convergence of the flow,we have to give the uniform interior C -estimates on u which is independent of t >
0. Finallythe asymptotic boundary behavior near the boundary as t → ∞ is established, which impliesthat the limit function is a solution to the generalized Loewner-Nirenberg problem. Many of thebarrier functions in these estimates can be viewed as a parabolic version of those in [7] and [5].This flow approach works well for the Loewner-Nirenberg problem of more general nonlineareuqations in [5]. Corollary 1.4.
Assume ( M n , g ) is a compact manifold with boundary of C ,α . Then there existsa sub-solution u to (1 . and a σ k -Ricci curvature flow g ( t ) = e u g starting from g = e u g andsatisfying (3 . and the Cauchy-Dirichlet condition (3 . − (3 . with some boundary data φ suchthat g ( t ) converges to g ∞ = e u ∞ g locally uniformly in C as t → + ∞ , where u ∞ is the uniquegeneralized Loewner-Nirenberg solution to (1 . i.e., u ∞ ( x ) → ∞ as x → ∂ M. Moreover, lim x → ∂ M ( u ∞ ( x ) + log( r ( x ))) = . Proof.
As discussed in Section 2, by [16] there exists a metric in the conformal class [ g ] of C ,α , which is still denoted as g such that Ric g < − ( n − g . If M is a Euclidean domain, wecan alternatively just choose g to be the Euclidean metric. We then take g as the backgroundmetric. Now we choose a sub-solution u to (1 .
5) such that u satisfies (3 .
6) on the boundary.For instance, if ( M , g ) is a sub-domain in Euclidean space, we choose u to be either the globalsub-solution constructed in [7] (just take η ( s ) = s for the subsolution u in Section 2) for theconstants A and p large, or the solution to (1 .
5) with u = ∂ M obtained in [7] or [5].For general ( M , g ), with the background metric g satisfying Ric g < − ( n − g , we can eithertake u to be the solution to (3 .
1) with u = ∂ M obtained in [7] or [5], or use the globalsub-solution constructed in Section 2, or u = v − v ∈ C ,α ( M ) is any sub-solution of(1 .
5) and hence u is a strict sub-solution (with ” > ” instead of ” = ” in (3 . φ ∈ C , ( ∂ M × [0 , ∞ )) satisfying the compatible condition (3 .
4) at t = φ ∈ C + α, + α ( ∂ M × [0 , T ]) for any T > φ t ≥ ∂ M × [0 , ∞ ) and φ ( x , t ) ≥ ξ ( t ) on ∂ M × [ T , ∞ ) for some T >
0, where ξ ( t ) is a low-speed increasing function in Definition 1.2.Now we consider the solution to the Cauchy-Dirichlet boundary value problem (3 . − (3 . (cid:3) ne can easily adapt this approach to the convergence of a σ k -Ricci curvature flow to thesolution to the Dirichlet boundary value problem of (1 . Corollary 1.5.
Assume ( M n , g ) is a compact manifold with boundary of C ,α . Let ϕ ∈ C ,α ( ∂ M ) .Then there exists a sub-solution u to (1 . and a σ k -Ricci curvature flow g ( t ) = e u g startingfrom g = e u g and satisfying (3 . and some Cauchy-Dirichlet condition such that g ( t ) con-verges to g ∞ = e u ∞ g uniformly in C as t → + ∞ , where u ∞ is the unique solution to (1 . suchthat u ∞ = ϕ on ∂ M. Recently, in [4] the authors studied a more general fully nonlinear equations with less restric-tion on regularity and convexity on the nonlinear structures on smooth domains in Euclideanspace and obtained a unique continuous viscosity solution, which is locally Lipschitz in the in-terior and shares the same blowing up ratio with the solution to the Loewner-Nirenberg problemnear the boundary.The paper is organized as follows: In Section 2, we construct a global sub-solution in C ,α ( M )to the σ k -Ricci equation (1 . t , and establish the asymptotic behavior of the solution near the boundary (seeLemma 4.4) and prove Theorem 1.3. Finally we give a proof of Corollary 1.5. Acknowledgements.
The author would like to thank Professor Matthew Gursky for helpfuldiscussion and Professor Jiakun Liu for nice talks on nonlinear equations.2. A global subsolution to (1 . u ∈ C ,α ( M ) to the homogeneous Dirichlet boundaryvalue problem (1 . − (1 . M n , g ) be a compact Riemannianmanifold with boundary of C ,α . We extend the manifold to a new manifold with boundary M = M S ( ∂ M × [0 , ǫ ]) for some small constant ǫ with ∂ M = ∂ M × { } and extend g toa C ,α metric on M . One can construct a conformal metric h ∈ [ g ] of C ,α with Ric h < M , which always exists by the proof in [16]. Without loss of generality, we take h as thebackground metric and still denote h as g in M , with Ric g ≤ − δ g for some constant δ > M . In fact by scaling we assume Ric g ≤ − δ g with δ > ( n −
1) large in M .Notice that there exist two small constants 0 < ǫ < δ such that dist( x , ∂ M ) > ǫ + δ for x ∈ ∂ M , and also ǫ + δ is less than the injectivity radius of any point q in the tubularneighborhood of ∂ M Ω = { x ∈ M (cid:12)(cid:12)(cid:12) dist g ( x , ∂ M ) ≤ ǫ + δ } , with dist g ( x , ∂ M ) distance function to ∂ M , and moreover for x ∈ Ω , the distance dist g ( x , ∂ M )is realized by a unique point x ∈ ∂ M through a unique shortest geodesic connecting x and x ,which is orthogonal to ∂ M at x . For any x ∈ ∂ M , we pick up the point ¯ x ∈ M \ M on thegeodesic starting from x along the outer normal vector of ∂ M so that dist g ( x , ¯ x ) = ǫ . Wedefine the distance function r ( x ) = dist g ( x , ¯ x ) for x ∈ M . In particular, r ( x ) = ǫ and r is mooth for r ≤ δ + ǫ . It is clear that r ( x ) ≥ r ( x ) for any x ∈ M and the equality holds if andonly if x = x .Now for a fixed x ∈ ∂ M and the corresponding point ¯ x , we can choose the subsolution in thefollowing way: We let A > p > N = A [ − ( δ + r ( x )) − p + r ( x ) − p ]is large, and we define a convex function η ∈ C ( R ), so that η ( s ) = η ( A (2 δ + r ( x )) − p − r ( x ) − p )) for s ≤ A [(2 δ + r ( x )) − p − r ( x ) − p ] , and η ( s ) = s , for s ≥ A [( δ + r ( x )) − p − r ( x ) − p ] . It is clear that η ′ ( s ) ≥ η ′′ ( s ) ≥
0, for s ∈ R . Now we define u ( x ) = η ( A ( r ( x ) − p − r ( x ) − p )) , and hence u ∈ C ,α ( M ). We claim that we can choose uniform large constants A > p > x ∈ ∂ M so that u is a subsolution. First, we give the calculation ∇ u ( x ) = − Ap η ′ r − p − ∇ r , ∇ i ∇ j u ( x ) = A p η ′′ r − p − ∇ i r ∇ j r + p ( p + A η ′ r − p − ∇ i r ∇ j r − pA η ′ r − p − ∇ i ∇ j r = A p η ′′ r − p − ∇ i r ∇ j r + Apr − p − η ′ [( p + ∇ i r ∇ j r − r ∇ i ∇ j r ] , ∆ u ( x ) = A p η ′′ r − p − (cid:12)(cid:12)(cid:12) ∇ r (cid:12)(cid:12)(cid:12) + Ap ( p + η ′ r − p − (cid:12)(cid:12)(cid:12) ∇ r (cid:12)(cid:12)(cid:12) − Ap η ′ r − p − ∆ r = A p η ′′ r − p − + Apr − p − η ′ [( p + − r ∆ r ] , It is clear that for given δ > ǫ >
0, we can choose p > x ∈ M such that r ( x ) ≤ δ + r ( x ), we have that ( p + − r ∆ r >
0, where p > x ∈ ∂ M . In fact, we choose p > p + − r ∆ r ) g i j − ( n − r ∇ i ∇ j r ]is positive for x ∈ M such that r ( x ) ≤ r ( x ) ≤ δ + r ( x ). Therefore,( n − ∇ u ( x ) + ∆ u ( x ) g (2.1)is always non-negative on M . Since − Ric > δ g with some constant δ > ( n −
1) on M and | du ( x ) | g − du ( x ) ⊗ du ( x )is semi-positive, we have that for 0 ≤ s ≤ ∇ s u ( x ) ≡ sg − (1 − s ) Ric g + ( n − ∇ u ( x ) + ∆ u ( x ) + ( n − | du ( x ) | g − du ( x ) ⊗ du ( x )) ≥ ( s + (1 − s ) δ ) g ≥ g . By the definition of η , u ( x ) ≤ η ( A (( r ( x ) + δ ) − p − r ( x ) − p )) = A (( r ( x ) + δ ) − p − r ( x ) − p )for r ( x ) ≥ δ + r ( x ). Now A > p > A (( r ( x ) + δ ) − p − r ( x ) − p ) < −
12 log(( n − , and hence σ n ( g − ¯ ∇ s u ) ≥ σ n ( δ ji ) = > ¯ β n , n e nu (2.2) or x ∈ M with r ( x ) ≥ δ + r ( x ). On the other hand, for x ∈ M with r ( x ) ≤ δ + r ( x ), we have η ( A ( r ( x ) − p − r ( x ) − p )) = A ( r ( x ) − p − r ( x ) − p ) ,η ′ ( A ( r ( x ) − p − r ( x ) − p )) = ,η ′′ ( A ( r ( x ) − p − r ( x ) − p )) = , and hence, as discussed in [7], for A > p > ∇ u ( x ) > ( n − g , (2.3)for x ∈ M with r ( x ) ≤ δ + r ( x ), where the term (2 .
1) serves as the main controlling positive term.Since u ≤
0, we have u ∈ C ,α ( M ) is a subsolution to the σ n equation when r ( x ) ≤ δ + r ( x ) andhence a sub-solution on M by (2 . u ≤ ∂ M . Let S k = σ k ( ¯ ∇ u ) nk ! − for 1 ≤ k ≤ n .By Maclaurin’s inequality, S ≥ S ≥ .. ≥ S k k ≥ .. ≥ S n n , which implies that a subsolution to the σ n equation is a subsolution to the σ k equation for1 ≤ k ≤ n , while a supersolution of the σ equation such that ¯ ∇ u ∈ Γ + k is a supersolution tothe σ k equation for 1 ≤ k ≤ n . In particular, u serves as a subsolution to the σ k equations anda uniform lower bound of the solutions to the homogeneous Dirichlet boundary value problemfor 1 ≤ k ≤ n . Moreover, by (2 . .
3) and the fact u ≤ ∂ M , we have σ k ( ¯ ∇ u ) > ¯ β k , n e ku (2.4)on M . Recall that A > p > x ∈ ∂ M . This proves the claim .Therefore, we have constructed a strict sub-solution u ∈ C ,α ( M ) to (1 .
5) and u ≤ M .3. A priori estimates for the σ k -R icci curvature flow On a compact Riemannian manifold ( M n , g ) with boundary ∂ M of C ,α . We denote M ◦ theinterior of M . If ( M , g ) is a bounded domain in the Euclidean space R n , we choose the naturalextension ( M , g ) which is a small tubular neighborhood of M in R n , and the global subsolutionused in [7] has no singularity in M . For general compact Riemannian manifold ( M n , g ) withboundary, with the extension ( M , g ) in Section 2, we choose g (and hence g on M ) to be theconformal metric which has − Ric g ≥ δ g with δ > n − k = , ..., n , we consider the Cauchy-Dirichlet problem of the σ k -Ricci curvature flow2 ku t = log( σ k ( ¯ ∇ u )) − log( ¯ β k , n ) − ku , on M × [0 , + ∞ ) , (3.1) u (cid:12)(cid:12)(cid:12) t = = u , (3.2) u (cid:12)(cid:12)(cid:12) ∂ M = φ, t ≥ , (3.3)where u ∈ C ,α ( M ) is a subsolution to the σ k -Ricci equation (1 . ∇ u is defined in (1 . φ ∈ C + α, + α ( ∂ M × [0 , T ]) for all T >
0, and moreover, φ satisfies φ t ≥ t ≥ φ ( t ) → + ∞ as t → + ∞ . To guarantee that the solution u to the Cauchy-Dirichlet problem of (3 .
1) satisfies ∈ C + α, + α ( M × [0 , T )) for some T >
0, we need the compatible condition(3.4) u ( x ) = φ ( x , , for x ∈ ∂ M , k φ t ( x , = log( σ k ( ¯ ∇ u )( x )) − log( ¯ β k , n ) − ku ( x ) , for x ∈ ∂ M , k φ tt ( x , = L ( v ( x )) , for x ∈ ∂ M , where the function v ∈ C ( M ) is v ( x ) ≡ k (log( σ k ( ¯ ∇ u )( x )) − log( ¯ β k , n ) − ku ( x ))(3.5)and L is the linear operator L ( ϕ ) = ¯ T i jk − σ k ( ¯ ∇ u ) [( n − ∇ i ∇ j ϕ + ∆ ϕ g i j + ( n − g km ∇ k u ∇ m ϕ g i j − ∇ i ϕ ∇ j u − ∇ i u ∇ j ϕ )] − k ϕ, for any ϕ ∈ C ( M ), where ¯ T i jk − is the ( k − ∇ u , which is positivedefinite. In order to find boundary data φ ∈ C + α, + α ( ∂ M × [0 , ∞ )) compatible with u such that φ t ≥ ∂ M × [0 , ∞ ), we need to assume that for the subsolution u ∈ C ,α ( M ), L ( v ( x )) ≥ x ∈ ∂ M such that v ( x ) =
0. We remark that sub-solutions u to (1 .
5) with thecondition (3 .
6) always exist on ( M , g ): It is clear that we do not need the condition (3 .
6) for asub-solution u which is strict on ∂ M i.e., σ k ( ¯ ∇ u ) > ¯ β k , n e ku for all x ∈ ∂ M . For instance, the global subsolution u we constructed in Section 2, by (2 . u = ϕ − C , with ϕ a sub-solution of (1 .
5) and C > u is a strict sub-solution of (1 .
5) on M . Also, if u ∈ C ,α ( M ) is a solution to (1 . v = M and hence (3 .
6) holds automatically. When u is a solution to (1 .
5) with u = ∂ M as obtained in [7] and [5], we can choose the boundary data φ = φ ( t ) ∈ C such that φ (0) = φ ′ (0) = φ ′′ (0) = φ ′ ( t ) ≥ t ≥
0. For a given constant T >
0, we call a function u ∈ C ( M × [0 , T )) a sub-solution ( super-solution ) of (3 .
1) if ¯ ∇ u ∈ Γ + k and u satisfies theinequality with ” ≤ ” (” ≥ ”) instead of ” = ” in (3 . . Lemma 3.1.
Let u and v be sub- and super- solutions to (3 . , with u ≤ v on ∂ M × [0 , T ) andM × { } , then we have u ≤ v on M × [0 , T ) .Proof. The proof is a modification of the maximum principle of σ k -Ricci equation in [7]. Weargue by contradiction. Let ξ = u − v . Assume that there exist 0 < t < T and x ∈ M ◦ such that ξ ( x , t ) = sup M × [0 , t ] ξ > . Then we have at ( x , t ), ˜ u t ≥ v t , ∇ ˜ u = ∇ v , ∇ ( v − ˜ u ) ≥ , nd hence ¯ ∇ ˜ u + V = ¯ ∇ v with V = ( n − ∇ ( v − ˜ u ) + ∆ ( v − ˜ u ) g ≥
0, which implies that σ k ( ¯ ∇ ˜ u ) ≤ σ k ( ¯ ∇ v ), and hence2 k ˜ u t − log( σ k ( ¯ ∇ ˜ u )) ≥ kv t − log( σ k ( ¯ ∇ v ))at ( x , t ). On the other hand, the function ˜ u = u − ξ ( x , t ) is a strict sub-solution to (3 .
1) on M × [0 , T ):2 k ˜ u t = ku t ≤ log( σ k ( ¯ ∇ u )) − log( ¯ β k , n ) − ku < log( σ k ( ¯ ∇ ˜ u )) − log( ¯ β k , n ) − k ˜ u . By the definition of sub- and super- solutions, we have at ( x , t ),2 k ˜ u t − log( σ k ( ¯ ∇ ˜ u )) < − log( ¯ β k , n ) − k ˜ u = − log( ¯ β k , n ) − kv ≤ kv t − log( σ k ( ¯ ∇ v )) , which is a contradiction. This proves the lemma. (cid:3) Based on the fact that the initial data u is a subsolution of (1 .
5) and the boundary data φ isincreasing in t , we have the monotonicity lemma. Lemma 3.2.
Assume that u ∈ C ( M ) is a subsolution to the σ k -Ricci equation (1 . , andu ∈ C , ( M × [0 , T )) is a solution to (3 . for some T > . Assume that u ( x , t ) = φ ( x , t ) for any ( x , t ) ∈ ∂ M × [0 , T ) and ∂∂ t φ ≥ on ∂ M × [0 , T ) . Then u t ≥ in M × [0 , T ) . In particular, u isincreasing along t ≥ . Moreover, we have upper bound estimates for u t on M × [0 , T ) .Proof. Let v = u t . We take derivative of t on both sides of the equation (3 .
1) to have2 kv t = σ k ( ¯ ∇ u ) ¯ T i jk − [( n − ∇ i ∇ j v + ∆ vg i j + ( n − g km u k v m g i j − v i u j − u i v j )] − kv , (3.7)where ¯ T i jk − is the ( k − ∇ u , which is positive definite since¯ ∇ u ∈ Γ + k . Recall that u is a subsolution of (1 . .
1) we have that v ( x , ≥ x ∈ M . Also, v ( x , t ) = φ t ( x , t ) ≥ x , t ) ∈ ∂ M × [0 , T ). We will use maximum principleto obtain that v ≥ M × [0 , T ). Otherwise, assume that there exists x ∈ M ◦ and t ∈ (0 , T )such that v ( x , t ) = inf M × [0 , t ] v < , then at ( x , t ), we have that v t ≤ , ∇ v = , ∇ v ≥ , v < , and hence v t ≤ , σ k ( ¯ ∇ u ) ¯ T i jk − [( n − ∇ i ∇ j v + ∆ vg i j + ( n − g km u k v m g i j − v i u j − u i v j )] − kv > , at ( x , t ), contradicting with the equation (3 . v = u t ≥ M × [0 , T ). Inparticular, u is a sub-solution to (1 .
5) for each t > v ( x , t ) = sup M × [0 , t ] v > x , t ) ∈ M ◦ × (0 , T ). Then at ( x , t ), v t ≥ , σ k ( ¯ ∇ u ) ¯ T i jk − [( n − ∇ i ∇ j v + ∆ vg i j + ( n − g km u k v m g i j − v i u j − u i v j )] − kv < , ontradicting with the equation (3 . .
1) at t =
0, we have v ( x , t ) = u t ( x , t ) ≤ max { k sup M [log( σ k ( ¯ ∇ u )) − log( ¯ β k , n ) − ku ] , sup ∂ M × [0 , t ] φ t } for any ( x , t ) ∈ M × [0 , T ). By integration, we have u ( x , t ) = u ( x ) + ˆ t u t ( x , s ) ds ≤ u ( x ) + t max { k sup M [log( σ k ( ¯ ∇ u )) − log( ¯ β k , n ) − ku ] , sup ∂ M × [0 , t ] φ t } , for any ( x , t ) ∈ M × [0 , T ); on the other hand, by monotonicity, u ( x , t ) ≥ u ( x ). Hence, we obtainthe upper and lower bound estimates for u on M × [0 , T ). (cid:3) We then give the boundary C estimates on u . Lemma 3.3.
Assume ( M n , g ) is a compact manifold with boundary of C ,α , and ( M , g ) is eithera compact domain in R n or with Ricci curvature Ric g ≤ − δ g for some δ ≥ ( n − . Letu ∈ C ( M × [0 , T )) be a solution to the Cauchy-Dirichlet problem (3 . − (3 . for some T > .Assume u ∈ C ,α ( M ) is a subsolution to (1 . satisfying (3 . at the points x ∈ ∂ M wherev ( x ) = . Also, assume φ ∈ C + α, + α ( ∂ M × [0 , T ]) for all T > , φ t ( x , t ) ≥ on ∂ M × [0 , + ∞ ) and φ satisfies the compatible condition (3 . with u . Then we have the boundary gradientestimates of u i.e., there exists a constant C = C ( T ) > such that |∇ u ( x , t ) | ≤ C (3.8) for ( x , t ) ∈ ∂ M × [0 , T ) .Proof. By the Dirichlet boundary condition, tangential derivatives of u on ∂ M × [0 , t ) is con-trolled by the tangential derivatives of the boundary data φ and hence, for the boundary gradientestimates of u , we only need to control | ∂∂ n u | with n the outer normal vector field of ∂ M .Since ¯ ∇ u ∈ Γ + k , we will show the lower bound of ∂∂ n u based on the control of sup M × [0 , T ) | u | as Guan’s argument in Lemma 5.2 in [5]. Indeed, we havetr( ¯ ∇ u ) = n − ∆ u + ( n − |∇ u | − n − R g ] ≥ , where R g ≤ Ric g ≤
0. Let v = e n − u . Then we have[ ∆ v − n − n − R g v ] ≥ . Let m = sup M × [0 , T ) | u | , which is bounded by the proof of Lemma 3.2. For any t >
0, let˜ v = ˜ v ( x , t ) be the solution to the Dirichlet boundary value problem of the linear elliptic equation ∆ ˜ v = n − n − R g e n − m , in M , ˜ v ( x , t ) = e n − φ ( x , t ) , p ∈ ∂ M . hen by continuity, for any T >
0, there exists a uniform constant C = C ( T ) >
0, such thatsup ( x , t ) ∈ ∂ M × [0 , T ] | ∂∂ n ˜ v | ≤ C ( T ) < + ∞ . For t < T , we have ∆ ˜ v ( x , t ) ≤ n − n − R g v ( x , t ) ≤ ∆ v ( x , t ) , ∀ x ∈ M , ˜ v ( x , t ) = v ( x , t ) , x ∈ ∂ M . By maximum principle, v ( x , t ) ≤ ˜ v ( x , t ) in M and since v ( x , t ) = ˜ v ( x , t ) for ( x , t ) ∈ ∂ M × [0 , T ),we have ∂∂ n v ≥ ∂∂ n ˜ v ≥ − C for some uniform constant C = C ( T ) > ∂ M × [0 , T ), and hence ∂∂ n u ≥ n − e − n − u ∂∂ n ˜ v ≥ − n − C ( T ) e − n − sup M × [0 , T | u | for ( x , t ) ∈ ∂ M × [0 , T ). This gives a uniform lower bound of ∂∂ n u on ∂ M × [0 , T ).Now we give upper bound estimates on ∂∂ n u . Let ( M , g ) be either a small tubular neighbor-hood of ( M , g ) in R n , or an extension of ( M , g ) as in Section 2 respectively. For any x ∈ ∂ M ,let ¯ x ∈ M \ M be as in Section 2 and r ( x ) be the distance function to ¯ x in M for x ∈ M . Let δ > δ < δ with δ > U = { x ∈ M , r ( x ) ≤ r ( x ) + δ } , with its boundary ∂ U = Γ S Γ where Γ = U T ∂ M and Γ = { x ∈ M (cid:12)(cid:12)(cid:12) r ( x ) = r ( x ) + δ } . Since 2 δ + r ( x ) is less than the injectivity radius at ¯ x , r ( x ) issmooth in U . For given T >
0, we extend φ to a C + α, + α function on U × [0 , T ] for any T > φ ( x , = u ( x ) for x ∈ U . Define the function u ( x , t ) = φ ( x , t ) + A ( 1 r ( x ) p − r ( x ) p ) , on U × [0 , T ], with two large constants A > p > A = A ( T ) and p = p ( T ) large so that u is a barrier function that controls the lower bound of u on U × [0 , T ]. Direct computations lead to u t = φ t , ∇ u = ∇ φ − Ap r − p − ∇ r , ∇ i ∇ j u = ∇ i ∇ j φ + Ap ( p + r − p − ∇ i r ∇ j r − Ap r − p − ∇ i ∇ j r ∆ u = ∆ φ + Ap ( p + r − p − (cid:12)(cid:12)(cid:12) ∇ r (cid:12)(cid:12)(cid:12) − Ap r − p − ∆ r = ∆ φ + Ap ( p + r − p − − Ap r − p − ∆ r . By continuity, there exist constants C > C = C ( T ) > |∇ r | + | ∆ r | ≤ C in U and |∇ φ | + |∇ φ | + | ∆ φ | ≤ C in U × [0 , T ]. We have the calculation( ¯ ∇ u ) i j = − Ric i j ( g ) + ( n − ∇ i ∇ j φ + Ap ( p + r − p − ∇ i r ∇ j r − Ap r − p − ∇ i ∇ j r ] + [ ∆ φ + Ap ( p + r − p − − Ap r − p − ∆ r ] g i j + ( n − |∇ u | g i j − ∇ i u ∇ j u ] . Since − Ric g ≥ ∇ i r ∇ j r ) and the last term are semi-positive, we have( ¯ ∇ u ) i j ≥ ( n − ∇ i ∇ j φ − Ap r − p − ∇ i ∇ j r ] + [ ∆ φ − Ap r − p − ∆ r ] g i j + Ap ( p + r − p − g i j , nd hence for any N > A >
0, there exists a constant p = p ( T , N , A ) >
0, such that for p > p , we have ( ¯ ∇ u ) i j ≥ N g i j on U × [0 , T ]. Let N ≥ ¯ β n n , n e U × [0 , T ] | φ t | + U × [0 , T ] | φ | . Then we have log( σ n ( ¯ ∇ u )) ≥ log( N n ) ≥ n φ t + log( ¯ β n , n ) + n φ ≥ nu t + log( ¯ β n , n ) + nu on U × [0 , T ]. Therefore, u is a subsolution of the σ n -Ricci curvature flow. By Maclaurin’sinequality, u is a subsolution of the σ k -Ricci curvature flow for any 1 ≤ k ≤ n . By definition, weknow that u ≤ u on Γ × [0 , T ). On Γ × [0 , T ), u and φ has uniform upper and lower bounds,and hence we can choose A and p large enough so that u < u on Γ × [0 , T ). Also, we have u ( x , ≤ φ ( x , = u ( x )for x ∈ U . By maximum principle in Lemma 3.1, we have that u ≥ u in U × [0 , T ). Since u ( x , t ) = φ ( x , t ) = u ( x , t ), we have ∂∂ n u ≤ ∂∂ n u at ( x , t ) for t ∈ [0 , T ], where n is the unit outer normal vector of ∂ M at x . Notice that theconstants used here can be chosen uniformly for all x ∈ ∂ M and hence, there exists a uniqueconstant m = m ( T ) >
0, such that ∂∂ n u ≤ m on ∂ M × [0 , T ). Therefore, we have the C estimates of u at points on ∂ M i.e., there exists a constant C = C ( T ) > |∇ u ( x , t ) | ≤ C for ( x , t ) ∈ ∂ M × [0 , T ). (cid:3) Now we give the C estimates of u on M × [0 , T ). Lemma 3.4.
Let ( M , g ) and u ∈ C ( M × [0 , T )) be as in Lemma 3.3. Then there exists a constantC = C ( T ) > such that |∇ u ( x , t ) | ≤ Cfor ( x , t ) ∈ M × [0 , T ) .Proof. The interior gradient estimate is relatively standard, and here we modify the argumentin [11] (see also [8]). By Lemma 3.2, there exist two constants −∞ < β < β < + ∞ dependingon T such that β ≤ u ≤ β on M × [0 , T ). We define a function ξ ( x , t ) = (1 + |∇ u | e η ( u ) , where η ( s ) = C ( C + s ) p s a function on s ∈ [ β , + ∞ ) with constants C > − β , C > p >
0, depending only on T , β and β , to be determined. Suppose that there exists x ∈ M ◦ and t ∈ (0 , T ) such that ξ ( x , t ) = sup M × [0 , t ] ξ. We take geodesic normal coordinates ( x , ..., x n ) centered at x ∈ M such that Γ mi j ( x ) = g i j ( x ) = δ i j . Then we have at ( x , t ), ξ x i = e η ( u ) [ u x a x i u x a + (1 + u x a u x a ) η ′ ( u ) u x i ] = , (3.9) ξ t = e η ( u ) [ u x a t u x a + (1 + u x a u x a η ′ ( u ) u t ] ≥ , (3.10)0 ≥ ξ x i x j = [ 12 ∂ ∂ x i x j g ab u x a u x b + u x a x i x j u x a + u x a x i u x a x j + η ′ ( u ) u x a x i u x a u x j + η ′ ( u ) u x a x j u x a u x i + (1 + |∇ u | )( η ′ ( u )) u x i u x j + (1 + |∇ u | ) η ′′ ( u ) u x i u x j + (1 + |∇ u | ) η ′ ( u ) u x i x j ] e η ( u ) = [ 12 ∂ ∂ x i x j g ab u x a u x b + u x a x i x j u x a + u x a x i u x a x j + (1 + |∇ u | )( η ′′ ( u ) − ( η ′ ( u )) ) u x i u x j + (1 + |∇ u | ) η ′ ( u ) u x i x j ] e η ( u ) , where the last identity is by (3 . Q i j ≡ σ k ( ¯ ∇ u ) (( n − T k − ) i j + g ab ( ¯ T k − ) ab g i j ) , is positive definite. Therefore, at ( x , t ),0 ≥ [ 1(1 + |∇ u | ) ( ¯ Q i j u x i x j x a u x a +
12 ¯ Q i j ∂ ∂ x i ∂ x j g ab u x a u x b + ¯ Q i j u x a x i u x a x j ) + ( η ′′ ( u ) − ( η ′ ( u )) ) ¯ Q i j u x i u x j + η ′ ( u ) ¯ Q i j u x i x j ] e η ( u ) . (3.11)By definition, at ( x , t ) we have¯ ∇ u = − Ric g + ( n − u x i x j + ∆ u δ i j − ( n − u x i u x j + ( n − |∇ u | δ i j , and hence by the identity ¯ T i j ( ¯ ∇ u ) i j = k σ k ( ¯ ∇ u ) and the equation (3 . Q i j u x i x j = σ k ( ¯ ∇ u ) [ ¯ T i j ( ¯ ∇ u ) i j + ¯ T i j (cid:0) Ric i j + ( n − u x i u x j − ( n − |∇ u | δ i j (cid:1) ] = σ k ( ¯ ∇ u ) [ k ¯ β k , n e ku t + ku + ¯ T i j (cid:0) Ric i j + ( n − u x i u x j − ( n − |∇ u | δ i j (cid:1) ] , (3.12)at ( x , t ). Now take derivative of x i on both sides of (3 . x , t ),2 ku tx i = σ k ( ¯ ∇ u ) ¯ T ab [ − ∂∂ x i Ric ab + ( n − u x a x b x i − ( n − ∂∂ x i Γ mab u x m + ( u x m x m x i − ∂∂ x i Γ cmm u x c ) g ab + ( n − u x m x i u x m g ab − u x a x i u x b − u x a u x b x i )] − ku x i , nd hence at ( x , t ), for 1 ≤ a ≤ n ,¯ Q i j u x i x j x a = k ( u tx a + u x a ) + σ k ( ¯ ∇ u ) ¯ T i j [( n − − u x m x a u x m g i j + u x i x a u x j + u x i u x j x a + ∂∂ x a Γ mi j u x m ) + ∂∂ x a Γ cmm u x c g i j + ∂∂ x a Ric i j ] . Now contracting this equation with ∇ u we have at ( x , t ),¯ Q i j u x i x j x a u x a = k ( u tx a u x a + u x a u x a ) + σ k ( ¯ ∇ u ) ¯ T i j [( n − − u x m x a u x m u x a g i j + u x i x a u x j u x a + ∂∂ x a Γ mi j u x m u x a ) + ∂∂ x a Γ cmm u x c u x a g i j + u x a ∂∂ x a Ric i j ] ≥ σ k ( ¯ ∇ u ) ¯ T i j [( n − (cid:0) + |∇ u | ) η ′ ( u )( u x a u x a g i j − u x i u x j ) + ∂∂ x a Γ mi j u x m u x a (cid:1) + ∂∂ x a Γ cmm u x c u x a g i j + u x a ∂∂ x a Ric i j ] + k ( u x a u x a − (1 + |∇ u | η ′ ( u ) u t ) , (3.13)where the last inequality is by (3 .
9) and (3 . .
12) and (3 .
13) to (3 . ≥ + |∇ u | ) [2 k ( |∇ u | − (1 + |∇ u | η ′ ( u ) u t ) + ¯ T i j σ k ( ¯ ∇ u ) ∂∂ x a Ric i j u x a + ¯ Q i j u x a x i u x a x j + ¯ Q i j R ia jb u x a u x b ] + n − σ k ( ¯ ∇ u ) ¯ T i j ( |∇ u | g i j − u x i u x j ) η ′ ( u ) + ( η ′′ − ( η ′ ) ) ¯ Q i j u x i u x j + η ′ σ k ( ¯ ∇ u ) [ k ¯ β k , n e ku t + ku + ¯ T i j ( Ric i j + ( n − u x i u x j − |∇ u | g i j ))] = ( n −
2) 1 σ k ( ¯ ∇ u ) ( η ′′ − ( η ′ ) − η ′ ) ¯ T i j u x i u x j + σ k ( ¯ ∇ u ) ( η ′′ − ( η ′ ) + ( n − η ′ ) |∇ u | X i ¯ T ii + η ′ k ¯ β k , n e ku + ku t σ k ( ¯ ∇ u ) + η ′ σ k ( ¯ ∇ u ) ¯ T i j Ric i j − k η ′ u t + + |∇ u | ) [2 k |∇ u | + ¯ T i j σ k ( ¯ ∇ u ) ∂∂ x a Ric i j u x a + ¯ Q i j u x a x i u x a x j + ¯ Q i j R ia jb u x a u x b ] . Recall that u and u t are uniformly bounded from above and blow on M × [0 , T ) by Lemma 3.2,and so is the term 1 σ k ( ¯ ∇ u ) = ¯ β − k , n e − ku t − ku . Since ¯ T k − and ¯ Q k − are positively definite, we have at ( x , t ),0 ≥ ( n − e ku σ k ( ¯ ∇ u ) ( η ′′ − ( η ′ ) − η ′ ) ¯ T i j u x i u x j + e ku σ k ( ¯ ∇ u ) ( η ′′ − ( η ′ ) + ( n − η ′ ) |∇ u | X i ¯ T ii − C − C X i ¯ T ii , ith the constant C > T , sup ∂ M × [0 , T ) ( | φ | + | φ t | ), sup M log( σ k ( ¯ ∇ u )), sup M | u | ,sup M ( | Ric g | + |∇ Ric g | ) and sup β ≤ s ≤ β | η ′ ( s ) | . By the definition of η , we have η ′ >
0, and η ′′ − ( η ′ ) − η ′ = C p ( C + s ) p − [( p − − C p ( C + s ) p − ( C + s )] . For β ≤ s ≤ β , we choose C = − β , p > C > η ′′ − ( η ′ ) ≥ C p ,η ′′ − ( η ′ ) − η ′ ≥ , and hence at ( x , t ) |∇ u | X i ¯ T ii ≤ σ k ( ¯ ∇ u ) C pe ku ( C + C X i ¯ T ii ) = C p ¯ β k , n e ku t ( C + C X i ¯ T ii ) ≤ ¯ C (1 + X i ¯ T ii ) , where the constant ¯ C > T , sup ∂ M × [0 , T ) ( | φ | + | φ ′ | ), sup M log( σ k ( ¯ ∇ u )), sup M | u | ,sup M ( | Ric g | + |∇ Ric g | ) and sup β ≤ s ≤ β | η ′ ( s ) | . Recall that X i ¯ T ii = ( n − k + σ k − ( ¯ ∇ u ) ≥ ( n − k + nk − ! (cid:0) nk ! − σ k ( ¯ ∇ u ) (cid:1) k − k = ( n − k + nk − ! (cid:0) nk ! − ¯ β k , n e ku t + ku (cid:1) k − k ≥ C , (3.14)for some uniform constant C = C ( T ) >
0, where we have used the Maclaurin’s inequality andthe uniform lower bound of u and u t ≥
0. Therefore, |∇ u | ≤ ¯ C (1 + C ) . This combining with the boundary C estimates completes the proof of the gradient estimatesof u on M × [0 , T ). (cid:3) Now we consider the C estimates on u at the points on ∂ M × [0 , T ). Lemma 3.5.
Let ( M , g ) and u ∈ C ( M × [0 , T )) be as in Lemma 3.3. Then there exists a constantC = C ( T ) > such that |∇ u | ≤ Con ∂ M × [0 , T ) .Proof. We use the indices e i , e j to refer to the tangential vector fields on ∂ M and n the outernormal vector field. Notice that we have obtained the uniform boundssup ∂ M × [0 , T ) ( | u | + |∇ u | ) ≤ K , for some constant K > ∂ M × [0 , T ). By definition, we immediately have the control onthe second order tangential derivatives sup ∂ M × [0 , T ) |∇ i ∇ j u | ≤ C on ∂ M × [0 , T ) with some constant C > K and sup ∂ M × [0 , t ] ( | φ | + |∇ φ | + |∇ τ φ | )where ∇ τ φ means the second order tangential derivatives of φ on ∂ M . We extend φ to a function n C , ( U × [0 , + ∞ )) still denoted as φ such that φ ∈ C + α, + α ( M × [0 , T ]) for any T > φ ( x , = u ( x ) for x ∈ M .We now estimate the mixed second order derivatives |∇ n ∇ i u | with n the normal vector fieldon ∂ M . Let ( M , g ) be the extension of ( M , g ) as in Section 2. Let δ > ǫ > x ∈ ∂ M , let ¯ x be the point with respect to x as defined inSection 2. Define the exponential map Exp : ∂ M × [ − ǫ − δ, ǫ + δ ] → M such that Exp q ( s )is the point along the geodesic starting from q ∈ ∂ M in the normal direction of ∂ M of distance | s | to q . Here we take the inner direction to be positive i.e., Exp q ( s ) ∈ M ◦ when s >
0. Inparticular, ¯ x = Exp x ( − ǫ ). Notice that Exp : ∂ M × [ − ǫ − δ, ǫ + δ ] is a di ff eomorphism toits image. In fact we can choose ǫ + δ < ǫ where ǫ is strictly less than the lower bound ofinjectivity radius of each point in the thin ( ǫ + δ )-neighborhood Ω of ∂ M . We now use the Femicoordinate in a small neighborhood V x = B ǫ ( x ) of x in M : Let ( x , ..., x n − ) be a geodesicnormal coordinate centered at x on ( ∂ M , g (cid:12)(cid:12)(cid:12) ∂ M ). We take ( x ( q ) , ..., x n − ( q ) , x n ) as the coordinateof the point Exp q ( x n ) in V x . Define the distance function r ( x ) = dist( x , ¯ x ) for x ∈ M . Denote U = { x ∈ M (cid:12)(cid:12)(cid:12) r ( x ) ≤ δ + r ( x ) } , Γ = U T ∂ M and Γ = { x ∈ M (cid:12)(cid:12)(cid:12) r ( x ) = δ + r ( x ) } . By our choiceof the small constant ǫ + δ , we have Γ ⊆ V x and hence ∂∂ x i ( i < n ) is a tangential derivativeof ∂ M on Γ . It is clear that r ( x ) is smooth on U . The metric has the orthogonal decomposition g = d ( x n ) + g x n in U and we have Γ cab ( x ) = a , b , c ∈ { , , ..., n } . For i ∈ { , ..., n − } , taking derivative of ∂∂ x i on both sides of (3 .
1) we have0 = − ku tx i − ku x i + σ k ( ¯ ∇ u ) ¯ T ab [ −∇ i Ric ab + ( n − ∇ i ∇ a ∇ b u + ∇ i ∆ ug ab + n − (cid:0) ∇ i ∇ c u ∇ c ug ab − ∇ i ∇ a u ∇ b u (cid:1) ] . (3.15)Now we commute derivatives to have ∇ i ∇ a ∇ b u = ∇ a ∇ b u x i + Rm ∗ ∇ u , ∇ i ∆ u = ∆ u x i + Rm ∗ ∇ u , where the terms Rm ∗ ∇ u are contractions of some Riemannian curvature terms and ∇ u . Definethe linearized operator L acting on ϕ as L ( ϕ ) ≡ σ k ( ¯ ∇ u ) ¯ T ab [( n − ∇ a ∇ b ϕ + ∆ ϕ g ab + n − (cid:0) < ∇ ϕ, ∇ u > g ab − ∇ a ϕ ∇ b u (cid:1) ](3.16) − k ϕ t − k ϕ. Therefore, by (3 .
15) we have | L ( u x i ) | = σ k ( ¯ ∇ u ) | ¯ T ab ( −∇ i Ric ab + ( Rm ∗ ∇ u )) | ≤ C X i ¯ T ii (1 + |∇ u | ) ≤ C X i ¯ T ii , (3.17) or some constant C > M | Rm | , the lower bound of u t + u and the upper boundof |∇ u | on M × [0 , T ), which has been uniformly controlled. Recall that by (3 . X i ¯ T ii ≥ C for some uniform constant C = C ( T ) >
0, and hence direct calculation leads to the bound | L ( φ x i ) | ≤ C X i ¯ T ii + C ≤ C X i ¯ T ii , (3.18)on U × [0 , T ), where C > T , k , n ,sup M × [0 , T ) ( | u | + | u t | + |∇ u | ) and sup U × [0 , T ] ( | φ x i | + | φ tx i | + |∇ φ x i | + |∇ φ x i | ). Define the function v = u x i − φ x i in U × [0 , T ). Now by (3 .
17) and (3 .
18) we have | L ( v ) | ≤ C X i ¯ T ii , for some uniform constant C = C ( T ) >
0. Also, v = Γ .Now let ξ ( x ) = r ( x ) p − r ( x ) p for x ∈ U , where p > T to be determined. Following thecalculation in Section 2, we have that for p = p ( T ) > n − ∇ ξ + ∆ ξ g ≥ p r − p − g . Since ξ ≤ |∇ u | is uniformly bounded from above and u t + u is uniformly bounded from blow,we choose p = p ( T ) > L ( ξ ) ≥ β k , n e ku t + ku [ p r − p − − C |∇ u | |∇ ξ | ] X i ¯ T ii − k ξ ≥ C ( p r − p − − C pr − p − ) X i ¯ T ii ≥ p C r − p − X i ¯ T ii ≥ | L ( v ) | on U × [0 , T ) for some uniform constant C = C ( T ) >
0. Now we take p > ξ < −| v | on Γ × [0 , T ) and hence, ξ ≤ −| v | on ∂ U × [0 , T ). Recall that ξ ( x ) ≤ = v ( x , x ∈ M , we have by maximum principle, ± v ( x , t ) ≥ ξ ( x )for ( x , t ) ∈ U × [0 , T ). Since v ( x , t ) = ξ ( x ) =
0, we have for i = , ..., n − |∇ n u x i ( x , t ) | ≤ |∇ n φ x i ( x , t ) | + |∇ n v x i ( x , t ) | ≤ |∇ n φ x i ( x , t ) | + ∇ n ξ ( x ) ≤ C , or any ( x , t ) ∈ ∂ M × [0 , T ) with some uniform constant C = C ( T ) > x , t ) ∈ ∂ M × [0 , T ), where ∇ n is the outer normal derivative at x ∈ ∂ M . For thesecond order normal derivative ∇ n u , since tr( ¯ ∇ u ) ≥
0, i.e.2( n − ∆ u + ( n − n − |∇ u | − R g ≥ , by the estimates on the other second order derivatives, ∇ n u is bounded from below and we stillneed to derive an upper bound of ∇ n u . Orthogonally decompose the matrix ¯ ∇ u at x ∈ ∂ M innormal and tangential directions. By the previous estimates we have¯ ∇ u = ( n − u nn u nn g (cid:12)(cid:12)(cid:12) ∂ M ! + O (1)with the term | O (1) | ≤ C for some uniform constant C = C ( T ) > u nn → + ∞ , we have σ k ( ¯ ∇ u ) = ( u nn ) k ( Λ k , n + o (1)) → + ∞ , where Λ k , n is a positive constant. On the other hand, recall that0 < C ≤ σ k ( ¯ ∇ u ) = ¯ β k , n e ku t + ku ≤ C , for some uniform constant C = C ( T ) > M × [0 , T ) and hence, we have that there existsa uniform constant C = C ( T ) > ∇ n u ( x ) ≤ C . Notice that the constant C here isindependent of the choice of x ∈ ∂ M . This completes the boundary C estimates of u . (cid:3) Proposition 3.6.
Let ( M , g ) and u ∈ C ( M × [0 , T )) be as in Lemma 3.3. Then there exists aconstant C = C ( T ) > such that for any ( x , t ) ∈ M × [0 , T ) we have |∇ u ( x , t ) | g ≤ C . Proof.
The proof is a modification of Proposition 3.3 in [11], see also [8]. We have obtainedthe global C estimates and boundary C estimates on u . Now suppose the maximum of |∇ u | g is achieved at a point in the interior.Denote S ( T M ) the unit tangent bundle of ( M , g ). We define a function h : S ( T M ) × [0 , T ) → R , such that h ( x , e x , t ) = ( ∇ u + m |∇ u | g )( e x , e x ) , for any x ∈ M , t ∈ [0 , T ) and e x ∈ S T x M , with m > q , t ) ∈ M ◦ × [0 , T ) and a unit tangent vector e q ∈ S T q M such that h ( q , e q , t ) = sup S ( T M ) × [0 , t ] h . Notice that on S ( T M ) ⊆ S ( T M ) (here ( M , g ) is the extension of ( M , g ) as in Section 2),we can find a uniform constant C ′ > δ > x ∈ M and any e x ∈ T x M , e x can be extended to a unit vector field e on B δ ( x ) ⊆ M ◦ such that ∇ e ( x ) = |∇ e | ( x ) ≤ C ′ at this point x . Take the geodesic normal coordinates ( x , ..., x n )at q , and hence we have Γ cab ( q ) = g i j ( q ) = δ i j . By rotating, we assume ∇ u = u x i x j is iagonal at q and e q = ∂∂ x at ( q , t ). Let the unit vector field e = P i ξ i ∂∂ x i be the extension of e q on B δ ( q ) with ∇ e ( q ) = |∇ e | ( q ) ≤ C ′ . We have ξ ( q ) = , ξ i ( q ) = , i ≥ , and ∂∂ x i ξ j ( q ) = , i , j = , ..., n . It is clear that the fact ¯ ∇ u ∈ Γ + k and the uniform bound of |∇ u | on M × [0 , T ) imply that thereexists a uniform constant C > −∞ such that ∇ u > C at ( q , t ). Now we define a function ˜ h in asmall neighborhood U × [ t − ǫ, t + ǫ ] of ( q , t ) such that˜ h ( x , t ) = ( ∇ u + m |∇ u | g )( e , e ) = ξ i ξ j ( u x i x j − Γ ai j u x a ) + m |∇ u | . Since ˜ h achieves its maximum in U × [ t − ǫ, t ] at ( q , t ), we have that at ( q , t ), ∂∂ t ˜ h = u x x t + mu x a u x a t ≥ , (3.19) ˜ h x i = u x x x i − ∂∂ x i Γ a u x a + mu x a x i u x a = , (3.20) 0 ≥ ˜ h x i x j = u x x x i x j − ∂ ∂ x j ∂ x i Γ a u x a − ∂∂ x i Γ a u x a x j − ∂∂ x j Γ a u x a x i + m ∂ ∂ x i ∂ x j g ab u x a u x b + mu x a x i x j u x a + mu x a x i u x a x j + ∂ ∂ x i ∂ x j ξ a u x a x , where the last inequality means the Hessian of ˜ h is non-positive. Contracting the Hessian of ˜ h and the positively definite tensor ¯ Q i j ≡ σ k ( ¯ ∇ u ) (( n −
2) ¯ T i j + tr( ¯ T k − ) g i j ) we have at ( q , t )0 ≥ ¯ Q i j u x x x i x j − ¯ Q i j ∂ ∂ x j x i Γ a u x a − Q i j ∂∂ x j Γ a u x a x i + m ¯ Q i j ∂ ∂ x i ∂ x j g ab u x a u x b + m ¯ Q i j u x a x i x j u x a + m ¯ Q i j u x a x i u x a x j + Q i j ∂ ∂ x i ∂ x j ξ a u x a x . (3.21)Di ff erentiating equation (3 .
1) with respect to x a yields2 ku x a t + ku x a = σ k ( ¯ ∇ u ) ¯ T i j [ − ∇ a Ric i j + ( n − ∇ a ∇ i j u + ( ∆ u ) x a g i j + ( n − ∇ a ∇ b u ∇ b ug i j − ∇ a ∇ i u ∇ j u )] . Define the function F ( r i j ) = log( σ k ( r i j )) on Γ + k . Di ff erentiating (3 .
1) twice, we obtain2 k ∇ u t = ( ∂ F ∂ r ab ∂ r i j ) ∇ ( ¯ ∇ u ) ab ∇ ( ¯ ∇ u ) i j + σ k ( ¯ ∇ u ) ¯ T i j [ −∇ Ric i j + ( n − ∇ ∇ i j u + ∇ ( ∆ u ) g i j + n − < ∇ ∇ u , ∇ u > + ∇ ∇ a u ∇ ∇ a u ) g i j − ∇ ∇ i u ∇ j u − ∇ ∇ i u ∇ ∇ j u )] − k ∇ u ≤ σ k ( ¯ ∇ u ) ¯ T i j [2( n − (cid:0) ( < ∇ ∇ u , ∇ u > + ∇ ∇ a u ∇ ∇ a u ) g i j − ∇ ∇ i u ∇ j u − ∇ i u ∇ j u (cid:1) − ∇ Ric i j + ( n − ∇ ∇ i j u + ∇ ( ∆ u ) g i j ] − k ∇ u , ince F is concave on Γ + k . In particular, at ( q , t ) we rewrite these two derivatives as2 k ( u x a t + u x a ) = ¯ Q i j ( u x i x j x a − ∂∂ x a Γ bi j u x b ) + ¯ T i j σ k ( ¯ ∇ u ) [ −∇ a Ric i j + n − u x a x b u x b g i j − u x i x a u x j )] , (3.22)2 ku x x t ≤ ¯ T i j σ k ( ¯ ∇ u ) [2( n − (cid:0) ( u x x x a u x a − ∂∂ x Γ b a u x b u x a + u x x a u x x a ) δ i j − u x x x i u x j + ∂∂ x Γ b i u x b u x j − u x x i u x x j (cid:1) − ∇ Ric i j ] + ¯ Q i j [ u x i x j x x − ∂ ∂ ( x ) Γ ai j u x a − ∂∂ x Γ ai j u x x a − ∂∂ x Γ a i u x a x j ] − ku x x , and hence combining with (3 . ≥ ¯ Q i j ( ∂ ∂ ( x ) Γ ai j u x a + ∂∂ x Γ ai j u x a x + ∂∂ x Γ a i u x a x j − ∂ ∂ x j ∂ x i Γ a u x a − ∂∂ x j Γ a u x a x i ) − ¯ T i j σ k ( ¯ ∇ u ) [2( n − (cid:0) ( u x x x a u x a − ∂∂ x Γ b a u x b u x a + u x x a u x x a ) δ i j − u x x x i u x j − u x x i u x x j + ∂∂ x Γ b i u x b u x j ) − ∇ Ric i j ] + m ¯ Q i j ( ∂ ∂ x i ∂ x j g ab u x a u x b + ∂∂ x a Γ bi j u x b u x a + u x a x i u x a x j ) + mu x a (2 ku x a t + ku x a + ¯ T i j σ k ( ¯ ∇ u ) ( ∇ a Ric i j − n − u x a x b u x b g i j − u x a x i u x j ))) + k ( u x x t + u x x ) + Q i j ∂ ξ a ∂ x i ∂ x j u x a x . Therefore, by (3 .
19) and (3 .
20) we have0 ≥ ¯ Q i j ( ∂ ∂ ( x ) Γ ai j u x a + ∂∂ x Γ ai j u x a x + ∂∂ x Γ a i u x a x j − ∂ ∂ x j ∂ x i Γ a u x a − ∂∂ x j Γ a u x a x i ) − kmu x a u x a t + ku x x − ¯ T i j σ k ( ¯ ∇ u ) [2( n − (cid:0) ( ∂∂ x a Γ b u x b u x a − mu x b x a u x b u x a − ∂∂ x Γ b a u x b u x a + u x x a u x x a ) δ i j − ∂∂ x i Γ b u x b u x j + mu x b x i u x b u x j − u x x i u x x j + ∂∂ x Γ b i u x b u x j ) − ∇ Ric i j ] + m ¯ Q i j (2 R ia jb u x a u x b + u x a x i u x a x j ) + Q i j ∂ ξ a ∂ x i ∂ x j u x a x + mu x a (cid:0) ku x a t + ku x a + ¯ T i j σ k ( ¯ ∇ u ) ( ∇ a Ric i j − n − u x a x b u x b g i j − u x a x i u x j )) (cid:1) = ¯ Q i j (( ∇ R ai j − ∇ i R a j ) u x a − R a j u x a x i ) + ku x x + Q i j ( ∇ i j ξ a + R ai j ) u x a x − ¯ T i j σ k ( ¯ ∇ u ) [2( n − (cid:0) ( R b a u x b u x a − mu x b x a u x b u x a + u x x a u x x a ) δ i j − R b i u x b u x j + mu x b x i u x b u x j − u x x i u x x j ) − ∇ Ric i j ] + m ¯ Q i j (2 R ia jb u x a u x b + u x a x i u x a x j ) + mu x a (cid:0) ku x a + ¯ T i j σ k ( ¯ ∇ u ) ( ∇ a Ric i j − n − u x a x b u x b g i j − u x a x i u x j )) (cid:1) . y assumption, we have at ( q , t ), u x i x i ≤ u x x for i ≥ u x i x j = i , j . Recall that thereexists a unique C > −∞ on M × [0 , T ) such that u x x = ∇ u > C at ( q , t ) and hence, we have0 ≥ − C − Cu x x − (1 + m )( Cu x x + C ) X i ¯ T ii + σ k ( ¯ ∇ u ) [(2 m − n − u x x X i ¯ T ii + n − + m ) u x x i u x x j ¯ T i j ] ≥ − C − Cu x x − (1 + m )( Cu x x + C ) X i ¯ T ii + σ k ( ¯ ∇ u ) (2 m − n − u x x X i ¯ T ii , where C > M × [0 , T ) depending on k , n , C ′ , ( M , g ) andsup M × [0 , T ) ( | u | + | u t | + |∇ u | + | Rm | + |∇ Rm | + |∇ Ric | ) . Now take m to be a constant strictly larger than ( n − σ k ( ¯ ∇ u ) is uniformlybounded from above and below. On the other hand, by (3 . P i ¯ T ii > C for some uniformconstant C > M × [0 , T ), and hence we obtain that there exists a uniform constant C > M × [0 , T ), such that u x x ≤ C at ( q , t ). Therefore, combining with the boundary C estimates, we have that there exists auniform constant C > M × [0 , T ), such that |∇ u | ≤ C on M × [0 , T ). (cid:3) Remark. Here we give a way to extend the unit vector e q at q ∈ M ⊆ M in Proposition 3.6to a unit vector field e in a neighborhood of q with |∇ e | ( q ) ≤ C ′ for some C ′ > q ∈ M . Under the normal coordinates ( x , ..., x n ) in B δ ( q ) at q , Γ mi j (0) = g i j (0) = δ i j .Let ˜ e ( x ) = ∂∂ x for x ∈ B δ (0), where δ > q ∈ M in ( M , g ). Let e ( x ) ≡ ξ j ∂∂ x j = ˜ e ( x ) | ˜ e ( x ) | g for x ∈ B δ ( q ). Since ∇ i ˜ e j (cid:12)(cid:12)(cid:12) x = = ∂ ˜ e j ∂ x i = x = q ), we have ∇ i ξ j = ∂ i ( ˜ e j | ˜ e | g ) = ∂ i ˜ e j | ˜ e | − ∂ i ˜ e a ˜ e a ˜ e j | ˜ e | = , at the point q . Therefore, the extension ξ of e q in B δ ( q ) is a unit vector field with ∇ i ξ j ( q ) = C > M in ( M , g ), suchthat |∇ ξ ( q ) | ≤ C , for the extension e of e q defined above. . C onvergence of the σ k -R icci curvature flow Now we can prove the long time existence of the flow.
Theorem 4.1.
Assume ( M n , g ) is a compact manifold with boundary of C ,α , and ( M , g ) is eithera compact domain in R n or with Ricci curvature Ric g ≤ − δ g for some δ ≥ ( n − . Assumeu ∈ C ,α ( M ) is a subsolution to (1 . satisfying (3 . at the points x ∈ ∂ M where v ( x ) = . Also,assume φ ∈ C + α, + α ( ∂ M × [0 , T ]) for all T > , φ t ( x , t ) ≥ on ∂ M × [0 , + ∞ ) and φ satisfiesthe compatible condition (3 . with u . There exists a unique solution u ∈ C , ( M × [0 , + ∞ )) tothe Cauchy-Dirichlet problem (3 . − (3 . such that u ∈ C + α, + α ( M × [0 , T ]) for all T > , andthe equation (3 . is uniformly parabolic in t ∈ [0 , T ] for any T > .Proof. Since u is a subsolution to (1 . t =
0. By thecompatibility condition of φ and u , the implicit function theorem yields that there exists T > M × [0 , T ) and the Cauchy-Dirichlet problem has a uniquesolution u ∈ C , ( M × [0 , T )) such that u ∈ C + α, + α ( M × [0 , t ]) for any t ∈ (0 , T ). Recall that σ k ( ¯ ∇ u ) = ¯ β k , n e ku t + ku ≥ ¯ β k , n e ku , with the right hand side increasing by Lemma 3.2. Also, Lemma 3.2 gives the uniform upperand lower bounds of u on M × [0 , T ). By the a priori estimates in Lemma 3.4 and Proposition3.6, we have ¯ ∇ u ∈ Γ + k and the equation is uniformly parabolic, and hence Krylov Theorem forfully nonlinear parabolic equations yields uniform C ,α T ( M ) estimates on u with some constant0 < α T < t ∈ [0 , T ), see [9]. In turn the Schauder estimates yield uniform C + α, + α estimates on u in M × [0 , T ). Also, these a priori estimates apply to u on M × [0 , T ] forany T > T , and classical parabolic equationtheory applies to extend the flow to M × [0 , + ∞ ) and u ∈ C + α, + α ( M × [0 , T ]) for all T > (cid:3) To show the convergence of the flow, we establish the C and C interior estimates on u basedon the bound sup U × [0 , + ∞ ) | u | for any compact subset U ⊆ M ◦ . Lemma 4.2.
Assume u ∈ C , ( M × [0 , + ∞ )) is a solution to the Cauchy-Dirichlet boundaryvalue problem of the equation (1 . with u t ≥ . Assume that for any compact subset U ⊆ M ◦ ,there exists a constant C = C ( U ) > such that | u | ≤ C on U × [0 , + ∞ ) . Also, for some T > , we assume that there exists a constant C = C ( T ) > such that | u | + |∇ u | ≤ C ( T ) on M × [0 , T ] . Then for a point q ∈ M ◦ , there exists a constant C > depending on B r ( q ) ,C ( B r ( q )) and C ( T ) such that |∇ u | ≤ C on B r ( q ) × [0 , + ∞ ) , where r is the distance of q to ∂ M. roof. It is a modification of the interior estimates in [5]. For any T > T , we consider thefunction F ( x , t ) = µ ( x ) we f ( u ) on B r ( q ) × [0 , T ], where w = |∇ u | , and µ ∈ C ( B r ( q )) is a cut-o ff function such that µ = B r ( q ) , ≤ µ ≤ , |∇ µ | ≤ b µ , |∇ µ | ≤ b , (4.1)for some b > f ( u ) is to be determined later. By the assumption of thelemma, if F ( x , t ) achieves its maximum on B r ( q ) × [0 , T ] at a point ( x , t ) ∈ B r ( q ) × [0 , T ],then F ( x , t ) is uniformly bounded and hence |∇ u | ≤ C on B r ( q ) × [0 , T ] with a constant C > T . So from now on, we assume thatthere exists ( x , t ) ∈ B r ( q ) × ( T , T ] such that F ( x , t ) = sup B r ( q ) × [0 , T ] F . We choose the normal coordinate ( x , ..., x n ) at x . Then at ( x , t ), we have w t w + f ′ u t ≥ , (4.2) ∇ µµ + ∇ ww + f ′ ∇ u = , (4.3) ¯ T i j [ ∇ i ∇ j µµ − ∇ i µ ∇ j µµ + ∇ i ∇ j ww − ∇ i w ∇ j ww + f ′ ∇ i ∇ j u + f ′′ ∇ i u ∇ j u ] ≤ . (4.4)By (4 .
3) we have ¯ T i j ∇ i w ∇ j ww ≤ T i j ∇ i µ ∇ j µµ +
32 ( f ′ ) ¯ T i j ∇ i u ∇ j u , and hence plugging this inequality and the definition of w into (4 .
4) we have1 w ¯ T i j ∇ im u ∇ jm u + ¯ T i j ( ∇ i j µµ − ∇ i µ ∇ j µµ ) + w ¯ T i j ∇ i ∇ j ∇ m u ∇ m u + f ′ ¯ T i j ∇ i j u + ( f ′′ −
32 ( f ′ ) ) ¯ T i j ∇ i u ∇ j u ≤ . Dropping the non-negative first term, changing the order of derivatives for the third order deriv-ative term and by our choice of µ , we have at ( x , t ),1 w ¯ T i j ∇ m ∇ i ∇ j u ∇ m u + f ′ ¯ T i j ∇ i j u + ( f ′′ −
32 ( f ′ ) ) ¯ T i j ∇ i u ∇ j u ≤ ( C µ + C w − |∇ u | ) X i ¯ T ii = C ( 1 µ + X i ¯ T ii , or some uniform constant C > b and sup | Rm | on B r ( q ). Similar argumentyields 1 w ∇ m ∆ u ∇ m u + f ′ ∆ u + ( f ′′ −
32 ( f ′ ) ) |∇ u | ≤ C ( 1 µ + . Combining these two inequalities and the equation (3 . k ( u x i t u x i + |∇ u | ) σ k ( ¯ ∇ u ) − ¯ T ab ∇ i u ( −∇ i Ric ab + n − ∇ ic u ∇ c ug ab − ∇ ia u ∇ b u )) ≤ − w [( n − (cid:0) f ′ ¯ T i j ∇ i j u + ( f ′′ −
32 ( f ′ ) ) ¯ T i j ∇ i u ∇ j u (cid:1) + (cid:0) f ′ ∆ u + ( f ′′ −
32 ( f ′ ) ) |∇ u | (cid:1) X i ¯ T ii ] + w ( C µ + C ) X i ¯ T ii . Substituting (4 . .
3) and the following identity into this inequality¯ T ab ¯ ∇ ab u = ¯ T ab ( − Ric ab + ( n − ∇ ab u + ∆ ug ab + ( n − |∇ u | g ab − ∇ a u ∇ b u )) = k σ k ( ¯ ∇ u ) , we have at ( x , t ),2 k ( − f ′ u t w + |∇ u | ) σ k ( ¯ ∇ u ) − C |∇ u | X i ¯ T ii + n − w ¯ T i j [( ∇ c µ ∇ c u µ + f ′ |∇ u | ) g i j − ( ∇ i µ ∇ j u µ + f ′ ∇ i u ∇ j u )] ≤ − w [( n − f ′′ −
32 ( f ′ ) ) ¯ T i j ∇ i u ∇ j u + ( f ′′ −
32 ( f ′ ) ) |∇ u | X i ¯ T ii ] − kw f ′ σ k ( ¯ ∇ u ) + f ′ w ¯ T ab ( − Ric ab + ( n − |∇ u | g ab − ∇ a u ∇ b u )) + w ( C µ + C ) X i ¯ T ii . If w ≤ x , t ), then we obtain the uniform upper bound of |∇ u | . So we assume w > w − on both sides of the inequality, and by (3 .
1) we obtain2 k ( − f ′ u t + + f ′ ) ¯ β k , n e k ( u t + u ) − C X i ¯ T ii + n −
2) ¯ T i j [ ∇ c µ ∇ c u µ g i j − ∇ i µ ∇ j u µ ] + [( n − f ′′ −
32 ( f ′ ) − f ′ ) ¯ T i j ∇ i u ∇ j u + ( f ′′ −
32 ( f ′ ) + ( n − f ′ ) |∇ u | X i ¯ T ii ] ≤ ( C µ + C ) X i ¯ T ii , at ( x , t ), with C > | Rm | + |∇ Ric | ) and b , and hence we have2 k ( − f ′ u t + + f ′ ) ¯ β k , n e k ( u t + u ) + [( n − f ′′ −
32 ( f ′ ) − f ′ ) ¯ T i j ∇ i u ∇ j u + ( f ′′ −
32 ( f ′ ) + ( n − f ′ − b ) |∇ u | X i ¯ T ii ] ≤ C (1 + b )( 1 µ + X i ¯ T ii or some C > n , sup( | Rm | + |∇ Ric | ) and b , where we have used the Cauchyinequality and the constant b > f ( u ) = (2 + u − inf B r ( q ) × [0 , + ∞ ) u ) − N for some constant N > − N − N − ≤ f ′ = − N (2 + u − inf B r ( q ) × [0 , + ∞ ) u ) − N − ≤ − N (2 + osc u ) − N − < , f ′′ −
32 ( f ′ ) + n − f ′ = N [( N + − N (2 + u − inf B r ( q ) × R + u ) − N − n − + u − inf B r ( q ) × R + u )] × (2 + u − inf B r ( q ) × R + u ) − N − ≥ N (2 + u − inf B r ( q ) × [0 , + ∞ ) u ) − N − [(1 − − N ) N + − n − + osc u )]where osc u = sup B r ( q ) × [0 , + ∞ ) ( u − inf B r ( q ) × [0 , + ∞ ) u ) ≤ B r q × [0 , ∞ ) | u | . Now we take N > f ′′ −
32 ( f ′ ) + n − f ′ > , and take b = ( n − N (2 + osc u ) − N − , and hence,2 kC ( − f ′ u t + + f ′ ) ¯ β k , n e k ( u t + u ) + |∇ u | X i ¯ T ii ≤ C (1 + b )( 1 µ + X i ¯ T ii (4.5)for some C > n , sup | u | , sup( | Rm | + |∇ Ric | ) and b . Notice that if u t < , since u t ≥
0, and u and f ′ ( u ) are uniformly bounded, we have for some uniform constant C > |∇ u | X i ¯ T ii ≤ C (1 + b )( 1 µ + X i ¯ T ii + C . On the other hand, by (3 . X i ¯ T ii ≥ ( n − k + nk − ! (cid:0) nk ! − ¯ β k , n e ku t + ku (cid:1) k − k ≥ C for a uniform C > | u | , and hence we have µ |∇ u | ≤ C at ( x , t ) for some uniform constant C > n , sup | u | , sup( | Rm | + |∇ Ric | ) and b ,independent of T . For the case u t ≥ at ( x , t ), the first term in (4 .
5) is positive and hence |∇ u | X i ¯ T ii ≤ C (1 + b )( 1 µ + X i ¯ T ii , nd again we have µ |∇ u | ≤ C at ( x , t ) for some uniform constant C > n , sup | u | , sup( | Rm | + |∇ Ric | ) and b ,independent of T . Therefore, by the arbitrary choice of T > T , F ( x , t ) ≤ F ( x , t ) ≤ Ce − N for ( x , t ) ∈ [0 , + ∞ ). In particular, |∇ u ( x , t ) | ≤ C for ( x , t ) ∈ B r ( q ) × [0 , + ∞ ), for some uniform constant C > n , sup B r ( q ) × [0 , + ∞ ) | u | ,sup M ( | Rm | + |∇ Ric | ), b and B r ( q ). Therefore, for any compact subsets U and U such that U ⊆ U ◦ ⊆ U ⊆ M ◦ , there exists a uniform constant C > U , sup U × [0 , + ∞ ) | u | andsup M ( | Rm | + |∇ Ric | ) such that |∇ u ( x , t ) | ≤ C + sup U × [0 , T ] |∇ u | for ( x , t ) ∈ U × [0 , + ∞ ). (cid:3) Based on the interior C estimates, the interior C estimates are relatively easy modificationsof the C estimates in Proposition 3.6. Lemma 4.3.
Assume u ∈ C , ( M × [0 , + ∞ )) is a solution to the Cauchy-Dirichlet boundaryvalue problem of the equation (1 . with u t ≥ . Assume that for any compact subset U ⊆ M ◦ ,there exists a constant C ( U ) > such that | u | ≤ C on U × [0 , + ∞ ) . Also, for some T > , we assume that there exists a constant C = C ( T ) > such that |∇ u | ≤ C ( T ) on M × [0 , T ] . Then for a point q ∈ M ◦ , there exists a constant C ′ > depending on B r ( q ) ,C ( B r ( q )) and sup B r ( q ) × [0 , ∞ ) |∇ u | such that |∇ u | ≤ C ′ on B r ( q ) × [0 , + ∞ ) , where r is the distance of q to ∂ M.Proof.
For any T > T , we consider the function H : S ( T M ) × [0 , T ) → R such that H ( x , e x , t ) = µ ( x ) h ( x , e x , t ) or x ∈ M , e x ∈ S T x M and t ≥
0, where h is defined in the proof of Proposition 3.6 and µ ∈ C ( B r ( q )) satisfies (4 .
1) for some constant b >
0. By continuity, there exists a point( q , t ) ∈ B r ( q ) × [0 , T ] and e q ∈ S T q M , such that H ( q , e q , t ) = sup S T M × [0 , T ] µ ( x ) h ( x , e x , t ) . If t ≤ T , then by assumption, |∇ u | and hence H are well controlled. Therefore, we assumethat t > T . The same as in Proposition 3.6, we choose the normal coordinates ( x , ..., x n ) at q so that e q = ∂∂ x and we extend e q to a unit vector field e = ξ i ∂∂ x i in the neighborhood of q in thesame way. We define the function˜ H ( x , t ) = H ( x , e ( x ) , t ) = µ ( x )˜ h ( x , t ) = µ ( x )( ξ i ξ j ∇ i ∇ j u + m |∇ u | )in a neighborhood of ( q , t ), for some constant m > q , t ), we have˜ h t = ∇ ∇ u t + m ∇ a u t ∇ a u ≥ , (4.6) ∇ µµ + ∇ ˜ h ˜ h = , (4.7) ¯ T i j [ ∇ i j µµ − ∇ i µ ∇ j µµ + ∇ i j ˜ h ˜ h − ∇ i ˜ h ∇ j ˜ h ˜ h + ∇ j ∇ i ξ a ∇ a u ] ≤ , ∆ µµ − |∇ µ | µ + ∆ ˜ h ˜ h − |∇ ˜ h | ˜ h + ∆ ξ a ∇ a u ≤ . Direct calculation and changing order of derivatives yield at ( q , t ), ∇ i ˜ h = ∇ ∇ ∇ i u + Rm ∗ ∇ u + m ∇ i ∇ a u ∇ a u , ∇ j ∇ i ˜ h = ∇ ∇ ∇ j ∇ i u + ∇ Rm ∗ ∇ u + Rm ∗ ∇ u + m ( ∇ a ∇ j ∇ i u ∇ a u + ∇ ja u ∇ ia u + Rm ∗ ∇ u ∗ ∇ u ) , and hence combining these inequalities at the maximum point ( q , t ) we have¯ T i j [( n − ∇ ∇ ∇ i ∇ j u + ∇ ∇ ∆ ug i j ] ≤ ¯ T i j [( n − ∇ ji ˜ h + ∆ ˜ hg i j ] − m [( n −
2) ¯ T i j ∇ a ∇ j ∇ i u ∇ a u + ∇ a ∆ u ∇ a u X i ¯ T ii ] − m [( n −
2) ¯ T i j ∇ ja u ∇ ia u + ∇ ba u ∇ ba u X i ¯ T ii ] + ( C + C |∇ u | ) X i ¯ T ii ≤ − ˜ h ¯ T i j [( n − ∇ i j µµ − ∇ i µ ∇ j µµ ) + ( ∆ µµ − |∇ µ | µ ) g i j ] + ( C + C |∇ u | ) X i ¯ T ii − m [( n −
2) ¯ T i j ∇ a ji u ∇ a u + ∇ a ∆ u ∇ a u X i ¯ T ii ] − m [( n −
2) ¯ T i j ∇ ja u ∇ ia u + ∇ ba u ∇ ba u X i ¯ T ii ] ≤ − m [( n −
2) ¯ T i j ∇ a ji u ∇ a u + ∇ a ∆ u ∇ a u X i ¯ T ii ] − m [( n −
2) ¯ T i j ∇ ja u ∇ ia u + ∇ ba u ∇ ba u X i ¯ T ii ] + C (1 + (1 + µ ) |∇ u | ) X i ¯ T ii , here C depends on sup | Rm | , b , sup B r ( q ) × [0 , ∞ ) |∇ u | and the uniform upper bound of |∇ e | ( q ) (seeProposition 3.6), and hence combining this inequality with the two inequalities (3 .
22) we have2 k ( ∇ u + ∇ u t ) σ k ( ¯ ∇ u ) − n −
2) ¯ T i j [( ∇ a u ∇ a u + ∇ a u ∇ a u ) g i j − ∇ i u ∇ j u − ∇ i u ∇ j u ] ≤ − km ( ∇ a u t ∇ a u + |∇ u | ) σ k ( ¯ ∇ u ) − m [( n −
2) ¯ T i j ∇ ja u ∇ ia u + ∇ ba u ∇ ba u X i ¯ T ii ] + C (1 + m + (1 + m + µ ) |∇ u | ) X i ¯ T ii . Plugging in (4 .
6) and (4 . k ∇ u σ k ( ¯ ∇ u ) − n −
2) ¯ T i j [ ∇ a u ∇ a ug i j − ∇ i u ∇ j u ] ≤ − km |∇ u | σ k ( ¯ ∇ u ) − m [( n −
2) ¯ T i j ∇ ja u ∇ ia u + ∇ ba u ∇ ba u X i ¯ T ii ] + C (1 + m + (1 + m + µ ) |∇ u | ) X i ¯ T ii . Since ∇ i u ( q , t ) = i ≥ ∇ u ( q , t ) ≥ ∇ ii u ( q , t )for i ≥
2, and hence we have2 k ( ∇ u + m |∇ u | ) σ k ( ¯ ∇ u ) + (2 m − n − ∇ u ∇ u X i ¯ T ii ≤ C (1 + m + n (1 + m + µ ) |∇ u | ) X i ¯ T ii . We take m large and use the equation (3 .
1) to obtain2 k ( ∇ u + m |∇ u | ) ¯ β k , n e k ( u + u t ) + ∇ u ∇ u X i ¯ T ii ≤ C (1 + (1 + µ ) |∇ u | ) X i ¯ T ii , for some uniform C > T , and hence if ∇ u ( q , t ) >
1, the first term in thisinequality is positive and since P i ¯ T ii is uniformly bounded from below by (3 . µ ∇ u ( q , t ) ≤ C , for some uniform constant C > T , and hence˜ H ≤ C in B r ( q ) × [0 , T ] with C > T ; while if ∇ u ( q , t ) ≤
1, we trivially have theuniform upper bound of ˜ H by its definition and the bound of |∇ u | on B r ( q ) × [0 , ∞ ). By thearbitrary choice of T > T , ˜ H has a uniform upper bound on B r ( q ) × [0 , ∞ ). In particular, ∇ u ≤ C , n B r ( q ) × [0 , ∞ ). Since ¯ ∇ u ∈ Γ + k , and |∇ u | is uniformly bounded in B r ( q ), we have that thereexists a uniform constant α > −∞ such that ∆ u ≥ α, and hence |∇ u | ≤ n ( C + | α | ) , on B r ( q ) × [0 , ∞ ). This completes the proof of the lemma. (cid:3) Now we prove the convergence of the flow and the asymptotic behavior near the boundary as t → ∞ . Proof of Theorem 1.3.
Long time existence of the solution u has been obtained in Theorem4.1, and we only need the consider the convergence of u and its asymptotic behavior near theboundary as t → ∞ .First we establish the uniform upper bound estimates on u on any given compact subset of M ◦ . By the Maclaurin’s inequality, u is a subsolution to the σ -Ricci curvature flow (3 . σ -Ricci curvature flow in Lemma 3.1, to get the upper boundof u , it su ffi ces to find a super-solution to the scalar curvature equation i.e., (1 .
5) with k = .
7) near ∂ M . Direct application of Lemma 5.2 in [7], where a sequence of super-solutions to the scalar curvature equation on corresponding small geodesic balls blowing up onthe boundary was constructed, yields the upper bound of u :lim sup x → ∂ M [ u ( x , t ) + log( r ( x ))] ≤ , uniformly for all t >
0; and moreover, for any compact subset U ⊆ M ◦ , there exists a constant C > U such that u ( x , t ) ≤ C for all ( x , t ) ∈ U × [0 , + ∞ ). Here is an alternativeargument: by maximum principle for σ -Ricci curvature flow in Lemma 3.1, u ( x , t ) ≤ u LN ( x ) , for ( x , t ) ∈ M ◦ × [0 , ∞ ), where u LN is the solution to the Loewner-Nirenberg problem of theconstant scalar curvature equation on M . Recall that u LN ( x ) ≤ − log( r ( x )) + o (1) near the boundary , with o (1) → x → ∂ M , see in [15][14][1] for instance.By Lemma 3.2, u ( x , t ) is increasing along t > u ( x ) ≤ u ( x , t ) ≤ u LN ( x )for ( x , t ) ∈ M ◦ × [0 , + ∞ ). Or just use the super-solution to (1 .
5) on a small ball centered at x constructed in Lemma 5.2 in [7] instead of u LN . Therefore, u ( x , t ) converges as t → ∞ for any x ∈ M ◦ . By Lemma 4.2 and Lemma 4.3, we have that for any compact subsets U ⊆ U ⊆ M ◦ with U ⊆ U ◦ , there exists a constant C > |∇ u | + |∇ u | ≤ C in U × [0 , ∞ ) and hence, the equation (3 .
1) is uniformly parabolic and by (3 . u t has a uniformupper bound on U × [0 , ∞ ). By Krylov’s Theorem and the classical Schauder estimates, we ave that there exists a uniform constant C > U such that k u k C , ( U × [0 , ∞ )) ≤ C , and k u k C ,α ( U ) ≤ C , (4.8)for all t ≥
0. Since u increases and has uniform upper bound in U , by the Harnack inequality ofthe linear uniformly parabolic equation (3 .
7) for u t , we have v = u t → U as t → + ∞ . Therefore, u ( x , t ) → u ∞ ( x ) uniformly for x ∈ U as t → + ∞ . Bythe uniform bound (4 .
8) and the interpolation inequality, we have u ( x , t ) → u ∞ ( x )in C ( U ) as t → ∞ . By the arbitrary choice of the compact subset U ⊆ M ◦ , we have that u ∞ isa solution to (1 .
5) in M ◦ .Now we consider the lower bound of u near the boundary. Applying Lemma 4.4 to be provedlater, we have that there exist δ > T > u ( x , t ) ≥ − log( r ( x ) + ǫ ( t )) + w ( x )for x ∈ M with r ( x ) ≤ δ and t ≥ T , where w ( x ) ≤ w (cid:12)(cid:12)(cid:12) ∂ M = ǫ ( t ) → t → + ∞ .By the upper and lower bound estimates on u near the boundary, we have u ∞ ( x ) + log( r ( x )) → x → ∂ M . (cid:3) We will show the lower bound of the asymptotic behavior of u near the boundary as t → ∞ ,for which we need φ to increase not too slowly. Lemma 4.4.
Let ( M , g ) , u , φ , T > T and u be as in Theorem 1.3. Let r ( x ) be the distancefunction of x ∈ M to the boundary ∂ M. Then there exist δ > small and T > T , such thatu ( x , t ) ≥ − log( r ( x ) + ǫ ( t )) + w ( x ) for x ∈ M with r ( x ) ≤ δ and t ≥ T , where ǫ = ξ ( t ) − and w is a function of C where r ( x ) ≤ δ such that w ( x ) ≤ with w (cid:12)(cid:12)(cid:12) ∂ M = .Proof. Let δ > ∂ M × [0 , δ ] → M such that Exp q ( s ) ∈ M is the point on the geodesic starting from q ∈ ∂ M inthe direction of inner normal vector with distance s to q . δ is chosen small so that Exp is adi ff eomorphism to the image. Define U δ = { Exp q ( s ) (cid:12)(cid:12)(cid:12) ( q , s ) ∈ ∂ M × [0 , δ ] } . The metric has the orthogonal decomposition g = ds + g s , with g s the restriction of g on Σ s = { z ∈ M (cid:12)(cid:12)(cid:12) r ( z ) = s } for 0 ≤ s ≤ δ . Define the function u ( x , t ) = − log( r ( x ) + ǫ ( t )) + w ( x ) or ( x , t ) ∈ U δ × [ T , + ∞ ) where w ( x ) = A ( 1( r ( x ) + δ ) p − δ p )with constants A > p > δ > − ǫ ′ ( t ) ǫ ( t ) = ξ ′ ( t ) ≤ τ (4.9)for t ≥ T . Let ˜ r ( x , t ) = r ( x ) + ǫ ( t ). For any x ∈ U ◦ δ , let { e , ..., e n } be an orthonormal basis at x such that e = ∂∂ r . The same calculation as in Lemma 5.1 in [7] yields¯ ∇ u = − Ric g + ( n − ∇ w + ∆ wg + ( n − w ′ ) . . . + r ( n − g − n − rw ′ . . . − ˜ r (( n − ∇ r + ∆ rg ) . Recall that ∇ r and ∆ r are the second fundamental form and the mean curvature of Σ r ( x ) , whichare uniformly bounded by a constant γ ≥ U δ : γ g ≥ ( n − ∇ r + ∆ rg ≥ − γ g . We denote the bracketed term above on the right hand side as Φ . Taking δ + δ <
1, we have w ′ = − Ap ( r + δ ) − p − ≤ − Ap , and hence, 1˜ r Φ ≥ r ( n − g + ˜ r − γ n − Ap − γ . . . n − Ap − γ . Now we let δ < γ and choose T ′ > ǫ ( t ) < γ for t ≥ T ′ . There exists K > Ap > K , we have thatdet( 1˜ r Φ ) ≥ r n ( n − n (1 − γ ˜ rn − + n − Ap − γ n − r ) n − ≥ r n ( n − n (1 + Ap ˜ r ) . Recall that − Ric g ≥
0. For any large constant Λ >
0, there exists A > p > A > A and p ≥ p , ( n − ∇ w + ∆ wg ≥ Λ g , n U δ . Therefore, if we also assume Ap ≥ n τ , then we obtainlog(det( ¯ ∇ u )) − log( ¯ β n , n ) − nu ≥ log(det( 1˜ r Φ )) − log( ¯ β n , n ) − nu ≥ log (cid:0) ˜ r − n (1 + Ap ˜ r ) (cid:1) − nu ≥ log(1 + Ap ˜ r ) ≥ log(1 + n τ ˜ r ) , and hence for ˜ r ≤ (8 n τ ) − and t ≥ max { T , T ′ } , by (4 .
9) we havelog(det( ¯ ∇ u )) − log( ¯ β n , n ) − nu ≥ n τ ˜ r ≥ − n ǫ ′ ˜ r = nu t . (4.10)Since lim t →∞ ǫ ( t ) =
0, we take T ≥ max { T , T ′ } such that ǫ ( t ) ≤ (16 n τ ) − for t ≥ T and let δ < min { (16 n τ ) − , (20 γ ) − } . We will choose A and p large so that u gives a lower bound of u on U δ × [ T , ∞ ). Notice that ∂ U δ = Σ δ S ∂ M . By assumption we have u ( x , t ) = log( ξ ( t )) ≤ φ ( x , t )for ( x , t ) ∈ ∂ M × [ T , ∞ ). On Σ δ , since u is increasing, we have u ( x , t ) ≥ u ( x ). Notice thatthere exists A > A ≥ A and any p ≥
1, we have − log( δ ) + A (( δ + δ ) − p − δ − p ) < inf Σ δ u , and hence we have on Σ δ × [ T , ∞ ), u ≤ u . Finally, we consider the control on U δ × { T } . Since u ( · , T ) , u ( · , T ) ∈ C ( M ) and u ≤ u = φ on ∂ M × { T } , there exist A > p > A ≥ A and p ≥ p , we have u ≤ u on U δ × { T } .In summary, we assume Ap ≥ max { K , n τ } , p ≥ max { , p , p } , A ≥ max { A , A , A } ,δ + δ < , δ < min { (16 n τ ) − , (20 γ ) − } , and δ > ff eomorphism. Therefore, u is a sub-solution to (3 .
1) for k = n by (4 .
10) and hence a sub-solution to (3 .
1) for 1 ≤ k ≤ n on U δ × [ T , ∞ ) , by Maclaurin’sinequality; moreover, u ≤ u , on ( ∂ M [ Σ δ ) × [ T , ∞ ) [ U δ × { T } . Therefore, by the maximum principle in Lemma 3.1, we have u ( x , t ) ≥ u ( x , t ) = − log( r ( x ) + ǫ ( t )) + A (( r ( x ) + δ ) − p − δ − p )on U δ × [ T , ∞ ). (cid:3) roof of Corollary 1.5. The equation (1 .
5) is conformally covariant, and hence it is equivalentto consider the case when the background metric g is the Euclidean metric when ( M , g ) is adomain in the Euclidean space, while g ∈ C ,α is chosen to be a metric constructed in [16] (seeSection 2 in the present paper) such that Ric g < − ( n − g in the conformal class for a generalmanifold ( M , g ). Let u = u + min { , inf ∂ M ϕ } with u a sub-solution constructed in Section 2for A > p > M , g ) is a domain in Euclidean space one can just take u tobe the global sub-solution in [7] (just take the function η ( s ) = s for the sub-solution u in Section2) with A > p > u is a strict sub-solution near the boundary with u < ϕ on ∂ M and hence, we can construct the boundary data φ ∈ C + α, + α ( ∂ M × [0 , ∞ )) satisfyingthe compatible condition (3 .
4) at t = φ t ≥ ∂ M × [0 , ∞ ) and φ ( x , t ) → ϕ ( x )uniformly in C ,α ′ ( ∂ M ) as t → ∞ for some 0 < α ′ < . − (3 . u is a sub-solution tothe σ -Ricci curvature flow (3 . σ -Ricci equation (1 . u tothe Dirichlet boundary value problem with u = ϕ on ∂ M , see [17]. By Lemma 3.1 for the σ -Ricci curvature flow, we have u ( x , t ) ≤ u ( x ) for ( x , t ) ∈ M × [0 , ∞ ) and hence we have auniform upper bound of u . Also, the a priori C estimates from Lemma 3.3 to Proposition 3.6hold with uniform bound of k u ( · , t ) k C ( M ) independent of t >
0. By Theorem 4.1, we have thelong time existence of the unique solution u . Things are even better in this case: there exists auniform constant C > T > k u k C + α, + α ( M × [ T , T + ≤ C , (4.11)by Krylov’s Theorem and the standard Schauder estimates. Remark that here we do not needthe locally uniformly interior estimates.By (4 . t j → ∞ , such that u ( x , t j ) → u ∞ ( x ) in C ( M ) for some u ∞ ∈ C ,α ( M ) as t j → ∞ . By monotonicity of u , u ( x , t ) → u ∞ ( x ) uniformly for x ∈ M as t → ∞ . By (4 .
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Conformal geometry, contact geometry, and the calculus of variations , Duke Math. J., (2000), 283-316.G ang L i , D epartment of M athematics , S handong U niversity , J inan , S handong P rovince , C hina Email address : [email protected]@gmail.com