Conformal scalar curvature rigidity on Riemannian manifolds
aa r X i v : . [ m a t h . DG ] J un CONFORMAL SCALAR CURVATURE RIGIDITY ON RIEMANNIANMANIFOLDS
SEONGTAG KIM
Abstract.
Let ( M, ¯ g ) be an n -dimensional complete Riemannian manifold. In this paper,we considers the following conformal scalar curvature rigidity problem: Given a compactsmooth domain Ω with ∂ Ω, can one find a conformal metric g whose scalar curvature R [ g ] ≥ R [¯ g ] on Ω and the mean curvature H [ g ] ≥ H [¯ g ] on ∂ Ω with ¯ g = g on ∂ Ω? Weprove that ¯ g = g on some smooth domains in a general Riemannian manifold, which is anextension of the previous results given by Qing and Yuan, and Hang and Wang. Introduction
Let ( M, ¯ g ) be an n -dimensional complete Riemannian manifold and Ω be a smooth domainin ( M, g ) with smooth boundary ∂ Ω. Denote R [¯ g ] by the scalar curvature of ¯ g and H [¯ g ]the mean curvature of ∂ Ω. In this paper, we considers the following problem: Given acompact smooth domain Ω with ∂ Ω, can one find a conformal metric g whose scalar curvature R [ g ] ≥ R [¯ g ] on Ω and the mean curvature H [ g ] ≥ H [¯ g ] on ∂ Ω with ¯ g = g on ∂ Ω?For the conformal metric g of the given metric ( M, ¯ g ), scalar curvature R [ g ] and meancurvature H [ g ] on the boundary of the domain change in the following ways: g = e u ¯ g, R [ g ] = e − u ( R [¯ g ] − u ) and H [ g ] = e − u [ H [¯ g ] + ∂ ν u ] when n = 2 , (1) g = u n − ¯ g, R [ g ] = u − n +2 n − (cid:18) R [¯ g ] u − n − n − u (cid:19) and H [ g ] = u − nn − h H [¯ g ] + 2 n − ∂ ν u i when n ≥ . (2)Therefore the given condition R [ g ] ≥ R [¯ g ] on Ω is equivalent to R [¯ g ] e u ≤ ( R [¯ g ] − u ) when n = 2 and R [¯ g ] u n +2 n − ≤ (cid:18) R [¯ g ] u − n − n − u (cid:19) when n ≥ . (3)The condition H [ g ] ≥ H [¯ g ] with ¯ g = g on ∂ Ω is equivalent to ∂ ν u ≥ . (4)This problem is a conformal version of Min-Oo’s conjecture. Uniqueness and non-uniquenessof conformal metric with prescribed scalar curvature on Ω with the mean curvature conditionon ∂ Ω was studied by Escobar. He proved that on the annulus A a,b = { x ∈ R n | < a < | x |
Let ( M n , ¯ g, f ) be a complete n -dimensional static space with R ¯ g > ( n ≥ ). Assume Ω + ≡ { p ∈ M | f ( p ) > } is a pre-compact subset in M . Then, if ametric g ∈ [¯ g ] on M satisfies that • R [ g ] ≥ R [¯ g ] in Ω + , • g and ¯ g induced the same metric on ∂ Ω + , and • H [ g ] = H [¯ g ] on ∂ Ω + ,then g = ¯ g . For the proof of Theorem 1, they used the existence of the lapse functions f with the followingproperties:(5) − ∆ f − R [¯ g ] n − f = 0 and f > + and(6) ∇ f = 0 at ∂ Ω + = { x ∈ M n | f ( x ) = 0 } , where Ω + is the maximal subset where the conformal rigidity holds. The existence of lapsefunction comes from the vacuum static space ( M, g ) and (6) holds for f (see: [4, Theorem1]). In this paper, we extend the previous conformal scalar curvature rigidity results to thedomains in a general Riemannian manifold with the conformal invariant.2. Conformal Rigidity of Scalar curvature
Let (
M, g ) be a complete Riemannian manifold of dimension n ≥ R [ g ]. The Sobolev constant Q ( M, g ) of (
M, g ) and Q (Ω , g ) of a smooth domain Ω ⊂ ( M, g )are defined by Q ( M, g ) ≡ inf = u ∈ C ∞ ( M ) R M |∇ u | + ( n − n − R g u dV g (cid:0)R M | u | n/ ( n − dV g (cid:1) ( n − /n and Q (Ω , g ) ≡ inf = u ∈ C ∞ (Ω) R M |∇ u | + ( n − n − R g u dV g (cid:0)R M | u | n/ ( n − dV g (cid:1) ( n − /n . Note that Q ( M, g ) and Q (Ω , g ) are conformal invariant and Q (Ω , g ) ≥ Q ( M, g ). There aredomains in a complete Riemannian manifolds with positive Sobolev constant. For example,any simply connected domain in a complete locally conformally flat manifold has positiveSobolev constant [9]. It is known that for any smooth domain Ω ⊂ ( M, g ), Q (Ω , g ) ≤ ONFORMAL SCALAR CURVATURE RIGIDITY 3 Q ( S n , g ) where ( S n , g ) is the standard sphere (see [1]). Using Q (Ω , g ) and Q ( M, g ), weobtain conformal rigidity phenomena of scalar curvature. Let R + ( x ) = sup(0 , R [¯ g ]( x )). Theorem 2.
Let ( M n , ¯ g ) be a complete Riemannian n -manifold with scalar curvature R [¯ g ] ( n ≥ ) and Ω be a smooth domain in ( M n , ¯ g ) with positive Q (Ω , ¯ g ) > . Assume that ( n +2)4( n − h R Ω | R + | n ¯ g dV ¯ g i n < Q (Ω , ¯ g ) . Then, if a conformal metric g ∈ [¯ g ] on M satisfies that • R [ g ] ≥ R [¯ g ] in Ω , • g and ¯ g induced the same metric on ∂ Ω , and • H [ g ] ≥ H [¯ g ] on ∂ Ω ,then g = ¯ g .Proof. Since R [¯ g ] ≤ R [ g ] we have (3). Take v = u −
1, Ω = { x ∈ Ω | u ( x ) < } andΩ = { x ∈ Ω | u ( x ) ≥ } . We shall show that Ω = φ . If then, u > ∂ ν u ≥ ∂ Ω. However,this contradicts to the strong maximum principle since u > u = 1 on the ∂ Ω(see [7]). To show that Ω = φ , we let(7) A ( x ) = n − n − R [¯ g ] u ( x ) (cid:16) u ( x ) n − − (cid:17) u ( x ) − . The given conditions imply that(8) − ∆ v − A ( x ) v ≥ v = 0 on ∂ Ω and ∂ ν v = 0 on ∂ Ω. Note that A ( x ) ≤ n − R + [¯ g ] on Ω . Multiplying v on (8) on Ω ,(9) Z Ω |∇ v | − A ( x ) v dV ¯ g ≤ . We may consider v as a function defined on Ω by extending the domain. By using theSobolev constant Q (Ω , g ) of Ω, Q (Ω , ¯ g ) (cid:18)Z Ω | v | nn − dV ¯ g (cid:19) n − n ≤ Z Ω |∇ v | + ( n − n − R ¯ g v dV ¯ g ≤ Z Ω (cid:16) A ( x ) + ( n − n − R ¯ g (cid:17) v dV ¯ g ≤ (cid:16) n − n − n − (cid:17) Z Ω R +¯ g v dV ¯ g ≤ ( n + 2)4( n − Z Ω | R + | ¯ g v dV ¯ g ≤ ( n + 2)4( n − h Z Ω | R + | n ¯ g dV ¯ g i n h Z Ω | v | nn − dV ¯ g i n − n , (10)where (9) is used. Therefore ( n +2)4( n − h R Ω | R + | n ¯ g dV ¯ g i n < Q (Ω , g ) implies v ≡ . (cid:3) SEONGTAG KIM
Remark . Let Ω be a simply connected domain in a locally conformally flat manifold ( M, ¯ g ),then Q (Ω , g ) = Q ( S n , g ). If ( n +2)4( n − h R Ω | R + | n ¯ g dV ¯ g i n < Q ( S n , g ), then Theorem 2 holds forΩ. If ( M, g ) is locally conformally flat with constant positive scalar curvature, then therigidity holds for Ω with sufficiently small | Ω | . Remark . When ( M n , ¯ g ) is a compact Einstein manifold with positive scalar curvature, Q ( M n , ¯ g ) = ( n − n − R [¯ g ][ V ol ( M, g )] n [6, page 48]. Since Q ( M, ¯ g ) ≤ Q (Ω , ¯ g ), Theorem 2 holdsfor any smooth domain Ω with | Ω | < [ n − n +2 ] n [ V ol ( M, g )].For n ≥
2, let ( M n , ¯ g ) be a Riemannian space with positive scalar curvature R [¯ g ] > p there exist domain D ∋ p in a general manifold,on which conformal scalar curvature rigidity holds by applying the techniques of [7]. For adomain D ⊂ ( M, ¯ g ), we let R (¯ g, D ) = sup { p ∈ D } R [¯ g ]( p )and λ ( D ) be the 1-st nonzero eigenvalue of domain D with Dirichlet condition with respectto the metric ¯ g , i.e., λ ( D ) = inf u ∈ H R D |∇ u | dV ¯ g R D u dV ¯ g where u ∈ H = H , ( D ). It is known that for a given point p ∈ ( M, g ), we can find D ∋ p with sufficiently large λ ( D ). Theorem 5.
Let ( M n , ¯ g ) be a complete n -dimensional Riemannian space with R ¯ g > ( n ≥ ). Assume D is a smooth pre-compact subset in M with λ ( D ) > R (¯ g,D ) n − . Then, if ametric g ∈ [¯ g ] on M satisfies that • R [ g ] ≥ R [¯ g ] in D , • g and ¯ g induced the same metric on ∂D , and • H [ g ] = H [¯ g ] on ∂D ,then g = ¯ g .Proof. Let g = u n − ¯ g . To prove the rigidity on the domains in a general Riemannianmanifold, we construct a positive smooth function on a suitable domain D with the propertiessimilar to (5, 6). For this, we take any smooth domain D ⊂⊂ M with R (¯ g,D ) n − < λ ( D ) andthe eigenfunction u with ∆ u + λu = 0 on D . Take v = u −
1. Since u is positive on D ,we can express v = u β with some function β on D . From (8),0 ≥ ∆ v + A ( x ) v = △ ( u β ) + A ( x ) u β = u △ β + ∆ u β + ∇ u · ∇ β + A ( x ) u β = u △ β − λu β + ∇ u · ∇ β + A ( x ) u β. (11) ONFORMAL SCALAR CURVATURE RIGIDITY 5
Since u > D , 0 ≥△ β − λβ + ∇ u u · ∇ β + A ( x ) β ≥△ β + ( A ( x ) − λ ) β + ∇ u u · ∇ β. (12)Using L’hospital’s rule, β = 0 on ∂D . If there exists a minimum point p ∈ D with β ( p ) < ∇ β ( p ) = 0, then △ β ( p ) < A ( x ) ≤ λ ( D ) if v ≤
0. This contradicts to theMaximum principle (see [2]). Therefore v ≥ D . Then by the Maximum principle again,it does not satisfy the boundary condition ∂ ν u = 0 on ∂D if v is not identically zero on D . (cid:3) Note that on a standard hemisphere S + , R [¯ g ] = n ( n − λ ( S + ) = n (see [8]), whichprovides maximal domain. Any smaller domain than the hemisphere D ⊂⊂ S + satisfies λ ( D ) > n , on which Theorem 5 holds. References
1. T. Aubin, Some nonlinear problems in Riemannian geometry. Springer-Verlag, Berlin, 1998.2. W. Chen and C. Li,
Methods on Nonlinear Elliptic Equations , AIMS (2010)3. J. Escobar,
Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and aneigenvalue estimate
Vol 43, No7, Comm. Pure Appl. Math. (1990), 857-883.4. A. Fischer and J. Marsden,
Deformations of the scalar curvature , Vol.42, No.3 Duke MathematicalJournal (1975) 519 - 547.5. F. Hang and X. Wang,
Rigidity and non-rigidity results on the sphere , Comm. Anal. Geom. , (2006)91 - 106.6. E. Hebey, Sobolev spaces on Riemannian manifolds. Springer-Verlag, Berlin, 1996.7. J. Qing and W. Yuan On scalar curvature rigidity of Vacuum Static Spaces , Math. Ann. , (2016)1257–1277.8. R. Reilly,
Applications of the Hessian operator in a Riemannian manifold , Ind. Univ. Math. J. ,(1977) 459-472.9. R. Schoen, S. T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature , Invent. Math. , (1988), 47–71. Department of Mathematics Education, Inha University, Incheon 22212, Korea and De-partment of Mathematics, Princeton University, NJ 08544, USA
E-mail address ::