BOUNDARY EFFECT OF RICCI CURVATURE
|
|
- Hugh Reed
- 5 years ago
- Views:
Transcription
1 BOUNDARY EFFECT OF RICCI CURVATURE PENGZI MIAO AND XIAODONG WANG Abstract. On a compact Riemannian manifold with boundary, we study how Ricci curvature of the interior affects the geometry of the boundary. First we establish integral inequalities for functions defined solely on the boundary and apply them to obtain geometric inequalities involving the total mean curvature. Then we discuss related rigidity questions and prove Ricci curvature rigidity results for manifolds with boundary.. Introduction and statement of results In this paper, we consider the question how the Ricci curvature of a compact manifold with boundary affects the boundary geometry of the manifold. For the scalar curvature the same question is related to the quasi-local mass problem in general relativity. Indeed, much of the formulation of the results in this paper is motivated by that in [7, 20, 0]. We begin with integral inequalities that hold for functions solely defined on the boundary. For simplicity, all manifolds and functions in this paper are assumed to be smooth. Theorem.. Let Ω, g) be an n-dimensional, compact Riemannian manifold with nonempty boundary. Let K be a constant that is a lower bound of the Ricci curvature of g, i.e. Ric Kg. Let be the mean curvature of in Ω, g) with respect to the outward normal. Suppose > 0. Given any function η on, define η 2 η Aη) = dσ, Bη) = η dσ, [ ] η) 2 Cη) = II η, η) dσ, where, are the gradient, the Laplacian on respectively, II is the second fundamental form of and dσ is the volume form on. 200 Mathematics Subject Classification. Primary 53C20; Secondary 53C24. Research partially supported by Simons Foundation Collaboration Grant for Mathematicians #2805.
2 2 PENGZI MIAO AND XIAODONG WANG Then, for each nontrivial η, either ) 2 Bη).) Cη) Aη) Aη) or.2) 2 K Bη) Aη) Bη) Aη) ) 2 Cη) Aη). Remark.. If the term II η, η) were absent in Cη), then.) would always hold by ölder inequality. Remark.2. When Ω is the closure of a bounded domain in R 3, the functional Cη), up to a constant multiple of, is the 2nd variation 8π of the Wang-Yau quasi-local energy [2, 22]) at = Ω in R 3,, where R 3, is the 4-dimensional Minkowski spacetime. See [0, ] for details.) Remark.3. If has a component 0 on which II > 0, then.) always fails for an η which is a non-constant eigenfunction on 0 and zero elsewhere. In this case,.2) yields estimates on the first nonzero eigenvalue of 0. See Corollary 2. for details.) The conclusion of Theorem. is easily seen to be equivalent to a statement.3) II η, η)dσ η + tη)2 dσ for all constants t K. In Theorem 2. of Section 2, we prove a 2 more general version of.3) which allows 0. Interpreted this way, Theorem. and its generalization Theorem 2.) have natural applications to the total mean curvature of the boundary. We first state the case of nonnegative Ricci curvature. Theorem.2. Let Ω, g) be an n-dimensional, compact Riemannian manifold with nonnegative Ricci curvature, with connected boundary which has nonnegative mean curvature. Let X : R m be an isometric immersion of into some Euclidean space R m of dimension m n. Then.4) dσ 0 2 dσ, where 0 is the mean curvature vector of the immersion X, 0 is the length of 0, and = {x 0 x) 0}. Moreover, if equality in.4) holds, then
3 BOUNDARY EFFECT OF RICCI CURVATURE 3 a) = 0 identically on. b) Ω, g) is flat and X) lies in an n-dimensional plane in R m. c) Ω, g) is isometric to a domain in R n if X is an embedding. Remark.4. In light of the Nash imbedding theorem [2], the boundary always admits an isometric immersion into some Euclidean space. Therefore, Theorem.2 applies to any compact Riemannian manifold with nonnegative Ricci curvature, with mean convex boundary i.e. 0). One may compare Theorem.2 with the result in [7] in which a weaker curvature condition R 0 is assumed, where R is the scalar curvature, while a more stringent boundary condition is imposed. Remark.5. If Ω, g) has nonnegative Ricci curvature and nonempty mean convex boundary, it was shown in [7, 8] also cf. [5]) that Ω has at most two components, and Ω has two components only if Ω, g) is isometric to N I for a connected closed manifold N and an interval I. This is why we only consider connected boundary in Theorem.2. Remark.6. Theorem.2 generalizes [5, Proposition 2], which proves that b) and c) hold under a pointwise assumption 0. Indeed the proof in [5] can be easily adapted to prove our Theorem.2. Remark.7. If > 0 on, Theorem.2 implies dσ C where C > 0 is a constant depending only on the induced metric on and a positive lower bound of. Next, we give an analogous result for manifolds with positive Ricci curvature. Theorem.3. Let Ω, g) be an n-dimensional, compact Riemannian manifold with positive Ricci curvature, with connected boundary which has nonnegative mean curvature. Let k > 0 be a constant such that Ric n )kg. Suppose there exists an isometric immersion X : S m k, where Sm k is the sphere of dimension m n with constant sectional curvature k. Then.5) dσ < S n )2 k dσ, where S is the mean curvature vector of the immersion X. Like.4),.5) imposes constraints on the boundary mean curvature when the Ricci curvature of the interior has a positive lower bound. For instance, consider the standard hemisphere S n +, g S ) of dimension n.
4 4 PENGZI MIAO AND XIAODONG WANG Let Ω S n + be a smooth domain with connected boundary. It follows from Theorem.3 that there does not exist a metric g on Ω satisfying Ric n ), g T Ω = g S T Ω and S ) 2 + n 4 )2, where and S are the mean curvature of Ω in Ω, g) and Ω, g S ) respectively. This could be compared with the first step, i.e. [, Theorem 4], in the construction of the counterexample to Min-Oo s Conjecture, in which a metric on S n + is produced so that it satisfies R nn ), but the mean curvature of S n + is raised to be everywhere positive. One may also compare this with the Ricci curvature rigidity theorems in [6]. When a manifold has negative Ricci curvature somewhere, we have Theorem.4. Let Ω, g) be an n-dimensional, compact Riemannian manifold with boundary which has nonnegative mean curvature. Let k > 0 be a constant satisfying Ric n )kg. Suppose has a component 0 which admits an isometric immersion X = t, x,..., x n ) : 0 m k R m,, where R m, is the m+)-dimensional Minkowski spacetime with m n and { } m m k = y 0, y,..., y m ) R m, y0 2 + yi 2 = k, y 0 > 0. Then.6) dσ + II t, t)dσ 0 0 { < 2 4 n )2 k + [ t 2 ] } 2 n )kt dσ, 0 where is the mean curvature vector of the immersion X into m k, 2 [ 4 n )2 k + t ] 2 2 n )kt 0 on 0, and 0 is the set consisting of x 0 such that 2 x) [ 4 n )2 k + t ] 2 2 n )kt x) > 0. Remark.8. The term 0 II t, t)dσ in.6) can be dropped if either II 0 or X 0 ) m k {t = t 0} for some constant t 0. For instance, this is the case if 0 can be isometrically immersed in a sphere. i=
5 BOUNDARY EFFECT OF RICCI CURVATURE 5 The fact that.5) and.6) are strict inequalities is due to the characterization of equality case in Theorem 2.. This leads naturally to rigidity questions in the context of Theorem.3 and.4. We have the following two related results. Theorem.5. Let Ω, g) be an n-dimensional, compact Riemannian manifold with boundary. Suppose Ric n ) g there exists an isometric immersion X : S m, where S m is a standard sphere of dimension m n II v, v) II S v, v), for any v T. ere II is the second fundamental form of in Ω, g) and II S is the vector-valued, second fundamental form of the immersion X. Then Ω, g) is spherical, i.e. having constant sectional curvature. Moreover if is simply connected, then Ω, g) is isometric to a domain in S n +. Theorem.6. Let Ω, g) be an n-dimensional, compact Riemannian manifold with boundary. Suppose Ric n ) g there exists an isometric immersion X : m, where m is a hyperbolic space of dimension m n II v, v) II v, v), for any v T. ere II is the second fundamental form of in Ω, g) and II is the vector-valued, second fundamental form of the immersion X. Then Ω, g) is hyperbolic, i.e. having constant sectional curvature. Moreover if is simply connected, then Ω, g) isometric to a domin in n. The rest of this paper is organized as follows. In Section 2, we prove Theorem 2. which implies Theorem.. In Section 3, we consider applications of Theorem 2. to the total boundary mean curvature and prove Theorem.2.4. In Section 4, we discuss the related rigidity question and prove Theorem.5 and A geometric Poincaré type inequality The main result of this section is the following geometric Poincaré type inequality for functions defined on the boundary of a compact Riemannian manifold. Theorem 2.. Let Ω, g) be an n-dimensional, compact Riemannian manifold with nonempty boundary. Suppose Ric n )kg and 0,
6 6 PENGZI MIAO AND XIAODONG WANG where Ric is the Ricci curvature of g, k is some constant, and is the mean curvature of in Ω, g) with respect to the outward normal. Suppose is not identically zero. Then 2.) II η, η)dσ \{ η+tη=0} η + tη)2 dσ for any nontrivial function η on and any constant t n )k. 2 ere II, ) is the second fundamental form of, and denote the gradient and the Laplacian on respectively. Moreover, equality in 2.) holds only if either k > 0, t = 0 and η is a constant; or k = t = 0 and η is the boundary value of some function u on Ω satisfying 2 u = 0. ere 2 denotes the essian on Ω, g). Remark 2.. The case k = 0, > 0 and t = 0 in 2.) was first proved in [] and is related to the second variation of Wang-Yau quasi-local energy [2, 22] at a closed 2-surface in R 3 R 3,. Proof. The basic tool we use is Reilly s formula [4]) [ 2 u 2 u) 2 + Ric u, u) ] dv 2.2) Ω = [ II u, u) 2 u) u ν ) ] 2 u dσ, ν which follows from integrating the Bochner formula. ere, dv denote the Laplacian, the volume form on Ω, g) respectively; ν is the outward unit normal to, and u is any function defined on Ω. Given any nontrivial η on and any constant λ nk, let u be the unique solution to { u + λu = 0 on Ω 2.3) u = η at. The fact that 2.3) has a unique solution in the case k > 0 follows from another theorem of Reilly [4, Theorem 4]) which states that the first Dirichlet eigenvalue λ of satisfies λ nk and λ = nk if and only if Ω, g) is isometric to a hemisphere in which case II is identically zero. Plug this u in 2.2), using the fact 2 u 2 = n u)2 + 2 u n u)g 2, λ u 2 dv = u 2 dv u u Ω Ω ν dσ,
7 BOUNDARY EFFECT OF RICCI CURVATURE 7 and the assumption Ric n )kg, we have 2.4) n [ II η, η) 2 ) nk λ) u 2 dv + Ω Ω η + n 2n λη Given any constant ɛ > 0, 2.4) implies 2.5) + 0. [ [ n 2 u + n λug 2 dv ) u ν II η, η) + + ɛ η + n ) 2 2n λη + ɛ η + n ) + ɛ 2n λη + + ɛ u ] 2 dσ ν ) nk λ) u 2 dv + 2 u + Ω Ω n λug 2 dv ) ] 2 u dσ. ν ) ] 2 u dσ ν Define η,λ = { x η + n λη = 0}. By Lebesgue s monotone 2n convergence theorem, we have 2.6) = lim ɛ 0 \ η,λ \ η,λ η + n + ɛ η + n 2n λη ) 2 2n λη dσ ) 2 dσ. Therefore, it follows from 2.5) and 2.6) that 2.7) II η, η)dσ \ η,λ η + n ) 2 2n λη dσ, which proves 2.) by setting t = n λ. 2n Next, suppose 2.8) II η, η)dσ = \ η,λ η + n ) 2 2n λη dσ. In particular, this shows 2.9) η + n ) 2n λη L 2 \ η,λ )
8 8 PENGZI MIAO AND XIAODONG WANG and the set {x \ η,λ x) = 0} has dσ-measure zero. ence, by 2.4) and 2.8), we have ) nk λ) u 2 dv + 2 u + n Ω Ω n λug 2 dv [ η + n ) \ η,λ 2n λη + u ] 2 2.0) dσ ν ) 2 u dσ, ν which implies η,λ 2.) nk λ) u = 0, 2 u + λug = 0 on Ω n and 2.2) η + u ν + n λη = 0 at. 2n If λ < nk, 2.) shows u is identically a constant, therefore λ = 0, k > 0 and η is a constant on. If λ = nk, 2.) shows 2.3) 2 u + kug = 0 on Ω, which implies 2.4) η + u ν + n )kη = 0 at. Comparing 2.4) to 2.2) with λ = nk, we have k = λ = 0. This completes the proof. When > 0, 2.) is simplified to II η, η)dσ η + tη)2 dσ. In this case, Theorem 2. is equivalent to a statement that, given any nontrivial η on, the quadratic form satisfies Q η t) := Aη)t 2 + 2Bη)t + Cη) 2.5) Q η t) 0, t n )k, 2
9 where 2.6) 2.7) BOUNDARY EFFECT OF RICCI CURVATURE 9 Aη) = Cη) = η 2 η dσ, Bη) = η dσ, [ ] η) 2 II η, η) dσ. Clearly 2.5) is equivalent to asserting that, for each fixed η, either 2.8) Bη) 2 Aη)Cη) or 2.9) n )k Bη) 2 Aη) Bη) Aη) ) 2 Cη) Aη). This explains how Theorem. follows from Theorem 2.. Next, we apply Theorem 2. to eigenvalue estimates on the boundary. Corollary 2.. Let Ω, g),, k,, II be given as in Theorem 2.. Suppose has a component 0 which is convex, i.e. II > 0 on 0. Let κ > 0 be a constant such that II κγ, where γ is the induced metric on 0. Let λ be a positive eigenvalue of on 0, γ). If κ 2 + 2k > 0, then [ 2.20) λ / 4 n ) κ ] 2 [ κ 2 + 2k, 4 n ) κ + ] ) 2 κ 2 + 2k. In particular, if k 0, the first nonzero eigenvalue λ 0 ) of 0, γ) satisfies 2.2) λ 0 ) 4 n ) [κ + κ 2 + 2k] 2. Proof. By defining η = 0 everywhere on \ 0, Theorem 2. implies II η, η)dσ 0 0 η + tη)2 dσ, for any η defined on 0 and any t n )k. Let Aη), Bη) and 2 Cη) be given in 2.6) and 2.7) with replaced by 0. Suppose η is a nonzero eigenfunction, i.e. η + λη = 0. Then ) ) Bη) 2 η Aη)Cη) = 0 2 dσ II η, η)dσ > 0. 0 Therefore, 2.9) holds, which shows 2.22) n )k λ 2 0 II η, η)dσ 0 η 2 dσ ) 2.
10 0 PENGZI MIAO AND XIAODONG WANG On the other hand, 2.23) II η, η)dσ κ η 2 dσ = κλ η 2 dσ and η 2.24) 0 2 dσ η 2 dσ. n )κ 0 ence, 2.22) 2.24) imply 2.25) 2 n )k λ κ n )λ. When κ 2 + 2k > 0, it follows from 2.25) that λ / 2 n [ κ κ 2 + 2k which completes the proof. ], 2 n [ κ + κ 2 + 2k] ), Remark 2.2. Corollary 2. is motivated by results in [3, 23]. If k = 0, 2.2) reduces to λ 0 ) n )κ 2 which is the estimate in [23]. 3. Application to total mean curvature In this section, we recall the statement of Theorem.2.4 and give their proof. We begin with the case of nonnegative Ricci curvature. Theorem 3.. Let Ω, g) be an n-dimensional, compact Riemannian manifold with nonnegative Ricci curvature, with connected boundary which has nonnegative mean curvature. Let X : R m be an isometric immersion of into some Euclidean space R m of dimension m n. Then 3.) dσ 0 2 dσ, where 0 is the mean curvature vector of the immersion X and is the set { 0 x) 0}. Moreover, if equality in 3.) holds, then a) = 0 identically on. b) Ω, g) is flat and X) lies in an n-dimensional plane in R m. c) Ω, g) is isometric to a domain in R n if X is an embedding. Proof. Since X is an isometric immersion, one has 3.2) X = 0. At any x, let {v α α =,..., n } T x be an orthonormal frame that diagonalizes II, i.e. IIv α, v β ) = δ αβ κ α where {κ,..., κ n }
11 BOUNDARY EFFECT OF RICCI CURVATURE are the principal curvature of in Ω, g) at x. Let {e,..., e m } denote the standard basis in R m and x i be the i-th component of X. Then m m n II x i, x i ) = IIv α, v β ) e i, v α e i, v β 3.3) i= = i= α,β= n κ α =. α= Set k = 0 in Theorem 2. and choose η = x i, t = 0 in 2.), we have 3.4) II x i, x i )dσ x i) 2 i dσ where i is the characteristic function of the set i = \ { x i = 0}. Summing 3.4) over i, using 3.2), 3.3) and the fact i, we have 3.5) dσ 0 2 dσ, which proves 3.). Next suppose 3.6) dσ = 0 2 dσ. Then it follows from 3.4) that 3.7) II x i, x i )dσ = x i) 2 i dσ, i. By the rigidity part of Theorem 2., there exist functions u i, i m, such that u i = x i at and 3.8) 2 u i = 0 on Ω. Moreover, by 2.2) or 2.4), we have 3.9) 0 + dφν) = 0 at, where Φ : Ω R m is a harmonic) map defined by Φ = u,..., u m ), dφ = du,..., du m ) is the associated tangent map, and ν is the unit outward normal to in Ω, g). We claim 3.0) dφν)x) 0, x. To see this, first consider a point y by 3.2)). At y, 3.9) implies 3.) dφν)y) 0 and dφν)y) X).
12 2 PENGZI MIAO AND XIAODONG WANG ence, the rank of dφ at y is n by 3.) and the fact Φ = X. On the other hand, 3.8) shows du i is parallel on Ω, i. Therefore, the rank of dφ equals n everywhere on Ω. In particular, this proves 3.0). By 3.9) and 3.0), we now have 3.2) {x x) 0} =. Thus 3.6) becomes 3.3) dσ = dφν) 2 dσ by 3.9). As is a nonempty open set in, 3.2) and 3.3) imply 3.4) dφν) z) = and dφν)z) X), z. It follows from 3.4), 3.9) and 3.2) that = 0 identically on. The rest of the claims now follows from [5, Proposition 2]. For completeness, we include the proof. By 3.4) and the fact Φ = X, one knows g = m i= du i du i at. As a result, g = m i= du i du i on Ω as both tensors are parallel. Clearly this shows Ω, g) is flat and Φ is an isometric immersion. Next, let v, w be any tangent vectors to Ω. 3.8) implies 3.5) 0 = vwφ) v wφ) = dφv) dφw)) dφ v w), where and denote the connection on Ω, g) and R m respectively. By definition, 3.5) shows Φ : Ω R m is totally geodesic. Therefore, ΦΩ) hence X)) lies in an n-dimensional plane in R m. Without losing generality, one can assume ΦΩ) R n. If X : R m is indeed an embedding, then X) = W where W is the closure of a bounded domain in R n. Since Φ is an immersion and Φ = X, one can show ΦΩ) W and Φ : Ω \ W \ W by checking that ΦΩ) \ W is both open and closed in R n \ W ). On the other hand, Φ being a local isometry implies Φ : Ω W is a covering map. Therefore, Φ is a homeomorphism, and hence an isometry between Ω and W. Theorem 3.2. Let Ω, g) be an n-dimensional, compact Riemannian manifold with positive Ricci curvature, with connected boundary which has nonnegative mean curvature. Let k > 0 be a constant such that Ric n )kg. Suppose there exists an isometric immersion X : S m k, where Sm k is the sphere of dimension m n with constant sectional curvature k. Then 3.6) dσ < S n )2 k dσ,
13 BOUNDARY EFFECT OF RICCI CURVATURE 3 where S is the mean curvature vector of the immersion X into S m k. Proof. We identify S m k with the sphere of radius k centered at the origin in R m+, i.e. S m k = { y,..., y m+ ) R m+ m+ i= y2 i = k} and view X = x,..., x m+ ) : S m k R m+ as an isometric immersion of into R m+. Let 0 denote the mean curvature vector of X : R m+, then 3.7) 0 = S + k 0, X X, where, is the inner product in R m+. Apply the fact 3.8) X = 0 and X, X = k, we have 3.9) have 0 = m+ i= xi x i + x i 2) = 0, X + n ). In Theorem 2., choose η = x i and t = n )k > 0 in 2.), we ) where i II x i, x i )dσ < is the characteristic function of the set i = \ { x i + 2 n )kx i = 0 [ x i + 2 n )kx i] 2 i dσ }. Summing 3.20) over i and using 3.7) 3.9), we have dσ < [ n )k 0, X + 4 ] n )2 k 2 X 2 dσ [ = S 2 + ] 4 n )2 k dσ, where we have also used 3.3). This proves 3.6). Theorem 3.3. Let Ω, g) be an n-dimensional, compact Riemannian manifold with boundary which has nonnegative mean curvature. Let k > 0 be a constant satisfying Ric n )kg. Suppose has a component 0 which admits an isometric immersion X = t, x,..., x n ) : 0 m k R m,,
14 4 PENGZI MIAO AND XIAODONG WANG where R m, is the m+)-dimensional Minkowski spacetime with m n and { } m m k = y 0, y,..., y m ) R m, y0 2 + yi 2 = k, y 0 > 0. Then 3.2) dσ + II t, t)dσ 0 0 { < 2 4 n )2 k + [ t 2 ] } 2 n )kt dσ, 0 where is the mean curvature vector of the immersion X into m k, 3.22) 2 [ 4 n )2 k + t ] 2 2 n )kt 0 on 0, and 0 is the set consisting of all x 0 such that 2 x) [ 4 n )2 k + t ] 2 2 n )kt x) > 0. Proof. Let M be the mean curvature vector of X : 0 R m,, then i= 3.23) M = k M, X X, and 3.24) X = M, X, X = k, where, = dy0 2 + m i= dy2 i is the Lorentzian product on R m,. By 3.24), we have m 0 = t t + t 2 ) + xi x i + x i 2) 3.25) = M, X + n ). At any x 0, let {v α α =,..., n } be an orthonormal frame in T x 0 such that IIv α, v β ) = δ αβ κ α where {κ,..., κ n } are the principal curvature of 0 in Ω, g) at x. We have i= n 3.26) II t, t) = κ α y0, v α 2 α=
15 BOUNDARY EFFECT OF RICCI CURVATURE 5 and 3.27) m II x i, x i ) = i= = n m ) κ α yi, v α 2 α= i= n κ α + y0, v α 2). α= Therefore, 3.28) dσ + 0 II t, t)dσ = 0 m i= 0 II x i, x i )dσ. Now choose η = x i on 0, η = 0 on \ 0, and t = n )k < 0 2 in Theorem 2., we have 3.29) 0 II x i, x i )dσ < 0i [ x i 2 i] 2 n )kx dσ 0i where 0i is the characteristic function of the set 0i = 0 \ { x i } 2 n )kx i = 0. Direct calculation using 3.23) 3.25) shows 3.30) m i= [ x i ] 2 2 n )kx i = 2 4 n )2 k + [ t 2 n )kt ] 2, which also proves 3.22). Summing 3.29) over i =,..., m and using 3.28) 3.30) together with the fact 0i 0, we have dσ + II t, t)dσ 0 0 { < 2 4 n )2 k + [ t 2 ] } 2 n )kt dσ. 0 This completes the proof.
16 6 PENGZI MIAO AND XIAODONG WANG 4. Rigidity results Inequalities 3.6) and 3.2) are not sharp in the context of Theorem 3.2 and Theorem 3.3. In these cases, one wonders if there exist sharp integral inequalities involving and S or ) which include a rigidity statement in the case of equality. In what follows, by scaling the metric, we assume Ric n )g or Ric n )g. In the latter case, the scalar curvature R of g satisfies R nn ). By the results in [20, 8, 9], there exists a sharp integral inequality relating and if the manifold Ω is spin and the boundary embeds isometrically in the hyperbolic space n as a convex hypersurface. On the other hand, the counterexample to Min-Oo s conjecture in [] shows that even the pointwise condition = S is not sufficient to guarantee rigidity if one only assumes R nn ). This gives rise to the following rigidity question: Question 4.. Let Ω, g) be an n-dimensional, compact Riemannian manifold with boundary. Let D S n + be a bounded domain with smooth boundary D, where S n + is the standard n-dimensional hemisphere. Suppose Ric n ) g there exists an isometry X : D S X, where, S are the mean curvature of, D in Ω, g), S n + respectively. Is Ω, g) isometric to D in S n +? At this stage, we do not know the answer to Question 4.. owever, it was shown in [6] that Question 4. has an affirmative answer if the assumption S X is replaced by a stronger assumption on the second fundamental forms. Theorem 4. [6]). Let Ω, g) be an n-dimensional, compact Riemannian manifold with boundary. Let D S n + be a bounded domain with smooth boundary D, where S n + is the standard n-dimensional hemisphere. Suppose Ric n ) g there exists an isometry X : D II II S X, where II, II S are the second fundamental form of, D in Ω, g), S n + respectively. Then Ω, g) is isometric to D in S n +. In the rest of this section, we prove Theorem.5 and.6, which are analogues of Theorem 4. when the boundary is only isometrically immersed in a sphere or in a hyperbolic space of higher dimension.
17 BOUNDARY EFFECT OF RICCI CURVATURE 7 Theorem 4.2. Let Ω, g) be an n-dimensional, compact Riemannian manifold with boundary. Suppose Ric n ) g there exists an isometric immersion X : S m, where S m is a standard sphere of dimension m n II v, v) II S v, v), for any v T. ere II is the second fundamental form of in Ω, g) and II S is the vector-valued, second fundamental form of the immersion X. Then Ω, g) is spherical, i.e. having constant sectional curvature. We divide the proof of Theorem 4.2 into a few steps. First, we fix some notations. Let denote the covariant derivative on S m, which is identified with the unit sphere centered at the origin in R m+. Given any α = α,..., α m+ ) S m, let F = F α be the restriction of the linear function α, x = α x + + α m+ x m+ to S m. The gradient of F on S m, denoted by F, is 4.) F x) = α α, x x, x S m. On, define f = F X. For simplicity, given any p, we let F be the component of F Xp)) normal to X T p ). Given v, w T p, recall that II S v, w) = X v)x w) ) is the component of X v)x w) normal to X T p ). We let S denote the mean curvature vector of X, which is the trace of II S. Lemma 4.. Along, one has f 2 + f ) F 2 =, 4.3) f + n ) f S, F = 0, 4.4) X f) F, n + II S f, f), n = 0. ere n is any vector that is normal to X) in S m. Proof. 4.2) follows from the fact F 2 + F 2 =. To show 4.3) and 4.4), we note that F on S m satisfies 4.5) 2 F = F g S, where g S is the standard metric on S m. 4.5) readily implies 4.6) 2 fv, w) II Sv, w), F = f v, w, v, w T,
18 8 PENGZI MIAO AND XIAODONG WANG where 2 denotes the essian on. Taking trace of 4.6) gives 4.3). 4.5) also implies, 0 = 2 F X f), n) = X f) F, n + II S f, f), n, which proves 4.4). The condition IIv, v) II S v, v), v T, implies S 0. By Reilly s theorem [4, Theorem 4]), to prove Theorem 4.2, it suffices to assume λ > n, where λ is the first Dirichlet eigenvalue of Ω, g). Under this assumption, we let u be the unique solution to { u + nu = 0 on Ω, 4.7) u = f at. On Ω, g), define φ = u 2 + u 2. A basic fact about φ is that it is subharmonic, which follows from 4.8) 2 φ = 2 u + ug 2 + Ric u, u) n ) u 2 0. On, define χ = u, where ν is the unit outward normal to in ν Ω, g). Then 4.9) φ = f 2 + χ 2 + f 2 = + χ 2 F 2 by 4.2) in Lemma 4.. Lemma 4.2. Along, the normal derivative of φ is given by φ 2 ν = f, χ II f, f) χ2 Proof. Direct calculation gives 4.0) φ 2 ν = 2 u u, ν) + fχ = 2 u f, ν) + χ [ 2 u ν, ν) + f ] S, F χ. = f, χ II f, f) + χ [ 2 u ν, ν) + f ]. By 4.2) and 4.7), at we have nf = u = f + χ + 2 u ν, ν) = n ) f + S, F + χ + 2 u ν, ν),
19 BOUNDARY EFFECT OF RICCI CURVATURE 9 which gives 4.) 2 u ν, ν) + f = S, F χ. The lemma follows from 4.0) and 4.). Proof of Theorem 4.2. Given any q, we choose α = Xq) S m. Then F Xq)) = 0 by 4.). ence, 4.2) fq) = 0, F q) = 0 and φq) = + χ 2 q). Consider p such that φp) = max Ω φ. By 4.9) and 4.2), 4.3) χ 2 p) F 2 p). Since f, φ p) = 0, taking n = F in 4.4), at p we have χ f, χ = X f) F, F 4.4) = II S f, f), F. If χ p) 0, it follows from Lemma 4.2, 4.3) and 4.4) that 4.5) φ 2 ν p) = II S χ f, f), F II f, f) S, F χ χ 2 ) II S f, f) II f, f) + S χ 2 0. If χ p) = 0, then Lemma 4.2 gives 4.6) φ 2 ν p) = f, χ II f, f). Moreover, F p) = 0 by 4.3). Taking the second order derivative of φ along f at p, we have 4.7) 0 2 f fφ))p) ) = f χ f, χ X f) F, F = f, χ 2 X f) F 2.
20 20 PENGZI MIAO AND XIAODONG WANG We claim, at p, 4.8) X f) F ) = II S f, f). To see this, take any v T, 4.5) and 4.9) imply X f) F, X v) 4.9) = 2 F X f), X v)) X f)x f), X v) = fvf) 2 f f, v) = 2 v f 2 + f 2) = 2 v φ χ 2). Clearly, v φ χ 2 ) varnishes at p. ence, X f) F )p) is normal to X T p ). This, together with 4.4), implies 4.8). Now it follows from 4.6), 4.7) and 4.8) that φ 2 ν p) II S f, f) II f, f) 0. By the strong maximum principle precisely the opf boundary point lemma), we conclude that φ must be a constant. ence, 2 u = ug by 4.8). Moreover, by 4.9) and 4.2), 4.20) χ 2 F 2 = c for some constant c 0. We have the following two cases: If c > 0, then χ 2 > F 2 0. This together with 4.5) and the fact φ is a constant implies S = = 0. Since II 0, we have II = 0 and II S = 0. Thus, X : S m is totally geodesic. ence X) lies in an n )-dimensional standard sphere S n S m. Since X : S n is an isometric immersion, we have X) = S n ; moreover X : S n is one-to-one as S n is simply connected. Therefore, is isometric to S n and is totally geodesic in Ω, g). By [6, Theorem 2], we conclude that Ω, g) is isometric to a standard hemisphere S n +. If c = 0, then φ = on and hence on Ω). In this case, along, u 2 = f 2 + χ 2 = f 2 + F 2 = F 2 X. In particular, u q) = 0 by 4.2). We also note that u q) = fq) = by the definition of F. Finally, we are in a position to show Ω, g) has constant sectional curvature. It suffices to assume Ω, g) is not isometric to S n +. Given any x in the interior of Ω, let q x such that distx, ) = distx, q x ), where dist, ) denotes the distance functional on Ω, g). Consider the
21 BOUNDARY EFFECT OF RICCI CURVATURE 2 function f and u constructed in the above proof by taking q = q x. Since Ω, g) is not isometric to S n +, the constant c in 4.20) must be 0, hence u satisfies 4.2) 2 u = ug, uq x ) = 0, uq x ) =. Let γ : [0, L] Ω, g) be the geodesic satisfying γ0) = q x, γl) = x and L = distx, ). Let ξ = γ 0). Given any constant l 0, L), there exists an open neighborhood W of ξ in S n such that the exponential map exp qx, ) is a diffeomorphism from 0, l) W R + S n onto its image in Ω, g). Now it is a standard fact that 4.2) implies 4.22) exp qx ) g) = dr 2 + sin r) 2 g S n on 0, l) W, where g S n is the standard metric on S n cf. [6] for details). Therefore, g has constant sectional curvature at γt) for any t < L. By continuity, g has constant sectional curvature at x. This completes the proof that Ω is spherical. As an application, we have the following rigidity result which is a spherical analogue of [5, Theorem ]. Corollary 4.. Let Ω, g) be an n-dimensional, compact Riemannian manifold with boundary. Suppose Ric n )g g has constant sectional curvature at every point on. If is simply connected with nonnegative second fundamental form II, then Ω, g) is isometric to a domain in S n +. Proof. Let R,,, ), denote the curvature tensor, the connection on respectively. By the Gauss equation and the Codazzi equation, R v, v 2, v 3, v 4 ) = v, v 3 v 2, v 4 v, v 4 v 2, v 3 + II v, v 3 ) II v 2, v 4 ) II v, v 4 ) II v 2, v 3 ), 0 = v II ) v 2, v 3 ) v 2 II ) v, v 3 ) where v,... v 4 T. As is simply connected, the fundamental theorem of hypersurfaces cf. [9]) implies there exists an isometric immersion Φ : S n with II as its second fundamental form. Since II 0, by a result of Do Carmo and Warner in [4], Φ is an embedding and Φ) is a convex hypersurface in a hemisphere S n +. Now apply Theorem 4.2, we conclude that Ω, g) has constant sectional curvature everywhere. Let D be the region bounded by Φ) in S n +. We glue Ω and S n \ D along the boundary via the isometric embedding Φ to get a closed manifold M, g). The fact that has the same second fundamental form in Ω, g) and S n and that both Ω, g) and S n have constant
22 22 PENGZI MIAO AND XIAODONG WANG sectional curvature imply that g is C 2 indeed smooth) across in M. ence, M, g) is a spherical space form whose volume is greater than half of S n because it contains a hemisphere). Therefore, M, g) is isometric to S n and Ω, g) is isometric to a domain in S n +. Remark 4.. One can also apply Theorem 4. in the above proof. Theorem.5 now follows from Theorem 4.2 and Corollary 4.. When Ric n ), similarly we have Theorem 4.3. Let Ω, g) be an n-dimensional, compact Riemannian manifold with boundary. Suppose Ric n ) g there is an isometric immersion X : m, where m is a hyperbolic space of dimension m n II v, v) II v, v), for any v T. ere II is the second fundamental form of in Ω, g) and II is the vector-valued, second fundamental form of the immersion X. Then Ω, g) is hyperbolic, i.e. having constant sectional curvature. The proof is parallel to that of Theorem 4.2. Let, denote the dot product on R m, and be the connection on m. Identify m with {x R m, x, x =, x 0 > 0}. For any α X) m, consider F x) = F α x) = α, x on m. Its gradient is F x) = α + α, x x. Thus F x) 2 = + F 2. Given any p, let F X p) be the component of F Xp) orthogonal to X T p ). On, define f = F X. Let u be the smooth solution to { u = nu on Ω u = f at and define χ = u. Then φ := ν u 2 u 2 is subharmonic as seen from 2 φ = 2 u ug 2 + Ric u, u) + n ) u 2. Similar to 4.9), we have φ = f 2 + χ 2 f 2 = + χ 2 F 2. By analyzing the normal derivative φ in the same way as in the proof ν of Theorem 4.2, we conclude by the strong maximum principle that φ is constant. Therefore, 2 u = ug and χ 2 F 2 is a nonnegative constant c along. If c > 0, it implies II = 0, which contradicts the fact m does not contain a closed totally geodesic submanifold.
23 BOUNDARY EFFECT OF RICCI CURVATURE 23 Therefore χ 2 F 2 = 0 at, which shows φ = on Ω. By the same argument as that of Theorem 4.2, we conclude that Ω, g) has constant sectional curvature. Corollary 4.2. Let Ω, g) be an n-dimensional, compact Riemannian manifold with boundary. Suppose Ric n )g g has constant sectional curvature at every point on. If is simply connected with nonnegative second fundamental form II, then Ω, g) is isometric to a domain in n. The proof is similar to that of Corollary 4.. Since g has constant sectional curvature along and is simply connected, by the Gauss and Codazzi equations and the fundamental theorem of hypersurfaces cf. [9]), there exists an isometric immersion Φ : n with II as its second fundamental form. Since II 0, by the remark in Section 5 of Do Carmo and Warner [4], Φ is an embedding and Φ) is a convex hypersurface in n. Apply Theorem 4.3 to Ω, g) and the embedding Φ, we conclude that g has constant sectional curvature everywhere on Ω. Now let D be the region bounded by Φ) in n. We glue Ω and n \ D along the boundary via the isometric embedding Φ to get a complete manifold M, g). The fact has the same second fundamental form in Ω, g) and n and both Ω, g) and n have constant sectional curvature imply that g is smooth across in M. ence, M, g) is a complete, hyperbolic manifold which, outside a compact set, is isometric to n minus a ball. We conclude that M, g) is isometric to n and Ω, g) is isometric to a domain in n. This final claim can be seen, for instance, by the following: Proposition 4.. Let M, g) be a complete, n-dimensional Riemannian manifold with Ric n ) g. Suppose that there exists a compact set K M s.t. M \ K is isometric to n \ B where B is a geometric ball. Then M is isometric to n. The Euclidean version of the result is well known e.g. it appears as an exercise in [3] several times). The hyperbolic case can be proved by similar methods. For lack of an exact reference, we outline a proof using Busemann functions. The main idea comes from Cai-Galloway [2]. We use the upper space model n = {x R n : x n > 0}. Without loss of generality we take o = 0,..., 0, ). For a sequence ɛ k 0, let S k M be the hypersurface corresponding to x n = ɛ k and q k the point corresponding to 0,..., 0, /ɛ k ). Let p k be the point on S k closest to
24 24 PENGZI MIAO AND XIAODONG WANG q k and γ k : [ a k, b k ] M be a minimizing geodesic from p k to q k s.t. γ k 0) K it is easy to see that any minimizing geodesics from p k to q k must intersect K). Passing to a subsequence ɛ k 0 if necessary, we can assume that γ k 0) ō and γ k converges to a geodesic line γ : R M. We consider the following generalized Busemann function βx) = lim k dō, S k ) dx, S k ). Then we have Claim: β n in the support sense. The crucial fact is that S k has constant mean curvature = n. The argument is the same as in Cai-Galloway [2]. We also have the standard Busemann function b associated with the ray γ [0, ) defined by bx) = lim k s dx, γs)). It is known that b n in the support sense. The rest of the proof is similar to Cai- Galloway [2] or the proof of the Cheeger-Gromoll splitting theorem. We have b+β) 0. By the triangle inequality one can show b+β 0. On the other hand b + β = 0 along γ. Therefore by the strong maximum principle b + β = 0. Then β = b and it is a smooth function with beta =. By the Bochner formula one can show that 2 β = g dβ dβ. From this identity one can show that M is isometric to the warped product R S n, dt 2 + e 2t h), where S, h) is a flat Riemannian manifold. It is then clear that S, h) must be the standard R n. This finishes the proof of Proposition 4.. References [] Brendle, S., Marques, F. C. and Neves, A., Deformations of the hemisphere that increase scalar curvature, Invent. Math ), [2] Cai, M. and Galloway, G. Boundaries of zero scalar curvature in the AdS/CFT correspondence. Adv. Theor. Math. Phys ), no. 6, ). [3] Choi,. I. and Wang, A. N., A first eigenvalue estimate for minimal hypersurfaces, J. Diff. Geom ), [4] do Carmo, M. P. and Warner, F. W. Rigidity and convexity of hypersurfaces in spheres. J. Differential Geometry 4 970), [5] ang, F. and Wang, X., Vanishing sectional curvature on the boundary and a conjecture of Schroeder and Strake, Pacific J. Math ), no. 2, [6] ang,f. and Wang,X., Rigidity theorems for compact manifolds with boundary and positive Ricci curvature, J. Geom. Anal. 9, 2009), [7] Ichida, R., Riemannian manifolds with compact boundary, Yokohama Math. J ) no. 2, [8] Kasue, A., Ricci curvature, geodesics and some geometric properties of Riemannian manifolds with boundary, J. Math. Soc. Japan ), no., 7 3.
25 BOUNDARY EFFECT OF RICCI CURVATURE 25 [9] Kwong, K.-K., On the positivity of a quasi-local mass in general dimensions, Comm. Anal. Geom ) no. 4, [0] Miao, P. and Tam, L.-F., On second variation of Wang-Yau quasi-local energy, Ann. enri Poincaré 5 204), no. 7, [] Miao, P. and Tam, L.-F. and Xie, N.-Q., Critical points of Wang-Yau quasilocal energy, Ann. enri Poincaré 2 20), no. 5, [2] Nash, J., The imbedding problem for Riemannian manifolds, Ann. of Math. 2) ), [3] Petersen, P., Riemannian geometry. Second edition. Graduate Texts in Mathematics, 7. Springer, New York, [4] Reilly, R. C., Applications of the essian operator in a Riemannian manifold, Indiana Univ. Math. J ), [5] Sacksteder, R., On hypersurfaces with no negative sectional curvatures, Amer. J. Math ), [6] Schroeder, V. and Strake, M., Rigidity of convex domains in manifolds with nonnegative Ricci and sectional curvature, Comment. Math. elv. 64 2) 989), [7] Shi, Y.-G. and Tam, L.-F., Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom ), [8] Shi, Y.-G. and Tam, L.-F., Rigidity of compact manifolds and positivity of quasi- local mass, Class. Quantum Grav ), [9] Spivak, M., A comprehensive introduction to differential geometry. Vol. IV. Second edition. Publish or Perish, Inc., Wilmington, Del., 979. [20] Wang, M.-T. and Yau, S.-T., A generalization of Liu-Yaus quasi-local mass, Comm. Anal. Geom., ), no. 2, [2] Wang, M.-T. and Yau, S.-T., Quasilocal mass in general relativity, Phys. Rev. Lett ), 020. [22] Wang, M. -T. and Yau, S.-T., Isometric embeddings into the Minkowski space and new quasi-local mass, Comm. Math. Phys ), no. 3, [23] Xia, C., Rigidity of compact manifolds withboundary and nonnegative Ricci curvature, Proc. Amer. Math. Soc ) no. 6, Pengzi Miao) Department of Mathematics, University of Miami, Coral Gables, FL 3346, USA. address: pengzim@math.miami.edu Xiaodong Wang) Department of Mathematics, Michigan State University, East Lansing, MI 48864, USA. address: xwang@math.msu.edu
Rigidity and Non-rigidity Results on the Sphere
Rigidity and Non-rigidity Results on the Sphere Fengbo Hang Xiaodong Wang Department of Mathematics Michigan State University Oct., 00 1 Introduction It is a simple consequence of the maximum principle
More informationCalculus of Variations
Calc. Var. DOI 10.1007/s00526-011-0402-2 Calculus of Variations Extension of a theorem of Shi and Tam Michael Eichmair Pengzi Miao Xiaodong Wang Received: 13 November 2009 / Accepted: 31 January 2011 Springer-Verlag
More informationarxiv: v1 [math.dg] 4 Sep 2009
SOME ESTIMATES OF WANG-YAU QUASILOCAL ENERGY arxiv:0909.0880v1 [math.dg] 4 Sep 2009 PENGZI MIAO 1, LUEN-FAI TAM 2 AND NAQING XIE 3 Abstract. Given a spacelike 2-surface in a spacetime N and a constant
More informationUNIQUENESS RESULTS ON SURFACES WITH BOUNDARY
UNIQUENESS RESULTS ON SURFACES WITH BOUNDARY XIAODONG WANG. Introduction The following theorem is proved by Bidaut-Veron and Veron [BVV]. Theorem. Let (M n, g) be a compact Riemannian manifold and u C
More informationTHE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS
THE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS AILANA FRASER AND JON WOLFSON Abstract. In this paper we study the topology of compact manifolds of positive isotropic
More informationSpacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds
Spacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds Alfonso Romero Departamento de Geometría y Topología Universidad de Granada 18071-Granada Web: http://www.ugr.es/
More informationarxiv:math/ v1 [math.dg] 7 Jun 2004
arxiv:math/46v [math.dg] 7 Jun 4 The First Dirichlet Eigenvalue and Li s Conjecture Jun LING Abstract We give a new estimate on the lower bound for the first Dirichlet eigenvalue for the compact manifolds
More informationPENGFEI GUAN AND XI SISI SHEN. Dedicated to Professor D. Phong on the occasion of his 60th birthday
A RIGIDITY THEORE FOR HYPERSURFACES IN HIGHER DIENSIONAL SPACE FORS PENGFEI GUAN AND XI SISI SHEN Dedicated to Professor D. Phong on the occasion of his 60th birthday Abstract. We prove a rigidity theorem
More informationLIST OF PUBLICATIONS. Mu-Tao Wang. March 2017
LIST OF PUBLICATIONS Mu-Tao Wang Publications March 2017 1. (with P.-K. Hung, J. Keller) Linear stability of Schwarzschild spacetime: the Cauchy problem of metric coefficients. arxiv: 1702.02843v2 2. (with
More informationTRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE. Luis Guijarro and Gerard Walschap
TRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE Luis Guijarro and Gerard Walschap Abstract. In this note, we examine the relationship between the twisting of a vector bundle ξ over a manifold
More informationERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX
ERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX JOHN LOFTIN, SHING-TUNG YAU, AND ERIC ZASLOW 1. Main result The purpose of this erratum is to correct an error in the proof of the main result
More informationDIAMETER, VOLUME, AND TOPOLOGY FOR POSITIVE RICCI CURVATURE
J. DIFFERENTIAL GEOMETRY 33(1991) 743-747 DIAMETER, VOLUME, AND TOPOLOGY FOR POSITIVE RICCI CURVATURE J.-H. ESCHENBURG Dedicated to Wilhelm Klingenberg on the occasion of his 65th birthday 1. Introduction
More informationAn introduction to General Relativity and the positive mass theorem
An introduction to General Relativity and the positive mass theorem National Center for Theoretical Sciences, Mathematics Division March 2 nd, 2007 Wen-ling Huang Department of Mathematics University of
More informationInequalities for the ADM-mass and capacity of asymptotically flat manifolds with minimal boundary
Inequalities for the ADM-mass and capacity of asymptotically flat manifolds with minimal boundary Fernando Schwartz Abstract. We present some recent developments involving inequalities for the ADM-mass
More informationHow curvature shapes space
How curvature shapes space Richard Schoen University of California, Irvine - Hopf Lecture, ETH, Zürich - October 30, 2017 The lecture will have three parts: Part 1: Heinz Hopf and Riemannian geometry Part
More informationEinstein H-umbilical submanifolds with parallel mean curvatures in complex space forms
Proceedings of The Eighth International Workshop on Diff. Geom. 8(2004) 73-79 Einstein H-umbilical submanifolds with parallel mean curvatures in complex space forms Setsuo Nagai Department of Mathematics,
More informationTHREE-MANIFOLDS OF CONSTANT VECTOR CURVATURE ONE
THREE-MANIFOLDS OF CONSTANT VECTOR CURVATURE ONE BENJAMIN SCHMIDT AND JON WOLFSON ABSTRACT. A Riemannian manifold has CVC(ɛ) if its sectional curvatures satisfy sec ε or sec ε pointwise, and if every tangent
More informationRigidity of outermost MOTS: the initial data version
Gen Relativ Gravit (2018) 50:32 https://doi.org/10.1007/s10714-018-2353-9 RESEARCH ARTICLE Rigidity of outermost MOTS: the initial data version Gregory J. Galloway 1 Received: 9 December 2017 / Accepted:
More informationQuasi-local mass and isometric embedding
Quasi-local mass and isometric embedding Mu-Tao Wang, Columbia University September 23, 2015, IHP Recent Advances in Mathematical General Relativity Joint work with Po-Ning Chen and Shing-Tung Yau. The
More informationHouston Journal of Mathematics c 2009 University of Houston Volume 35, No. 1, 2009
Houston Journal of Mathematics c 2009 University of Houston Volume 35, No. 1, 2009 ON THE GEOMETRY OF SPHERES WITH POSITIVE CURVATURE MENG WU AND YUNHUI WU Communicated by David Bao Abstract. For an n-dimensional
More informationAFFINE MAXIMAL HYPERSURFACES. Xu-Jia Wang. Centre for Mathematics and Its Applications The Australian National University
AFFINE MAXIMAL HYPERSURFACES Xu-Jia Wang Centre for Mathematics and Its Applications The Australian National University Abstract. This is a brief survey of recent works by Neil Trudinger and myself on
More informationON THE FOLIATION OF SPACE-TIME BY CONSTANT MEAN CURVATURE HYPERSURFACES
ON THE FOLIATION OF SPACE-TIME BY CONSTANT MEAN CURVATURE HYPERSURFACES CLAUS GERHARDT Abstract. We prove that the mean curvature τ of the slices given by a constant mean curvature foliation can be used
More informationAN INTEGRAL FORMULA IN KAHLER GEOMETRY WITH APPLICATIONS
AN INTEGRAL FORMULA IN KAHLER GEOMETRY WITH APPLICATIONS XIAODONG WANG Abstract. We establish an integral formula on a smooth, precompact domain in a Kahler manifold. We apply this formula to study holomorphic
More informationThe eigenvalue problem in Finsler geometry
The eigenvalue problem in Finsler geometry Qiaoling Xia Abstract. One of the fundamental problems is to study the eigenvalue problem for the differential operator in geometric analysis. In this article,
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationComplete spacelike hypersurfaces with positive r-th mean curvature in a semi-riemannian warped product
DOI: 10.1515/auom-2015-0041 An. Şt. Univ. Ovidius Constanţa Vol. 23(2),2015, 259 277 Complete spacelike hypersurfaces with positive r-th mean curvature in a semi-riemannian warped product Yaning Wang and
More informationEstimates in surfaces with positive constant Gauss curvature
Estimates in surfaces with positive constant Gauss curvature J. A. Gálvez A. Martínez Abstract We give optimal bounds of the height, curvature, area and enclosed volume of K-surfaces in R 3 bounding a
More informationRIEMANNIAN SUBMERSIONS NEED NOT PRESERVE POSITIVE RICCI CURVATURE
RIEMANNIAN SUBMERSIONS NEED NOT PRESERVE POSITIVE RICCI CURVATURE CURTIS PRO AND FREDERICK WILHELM Abstract. If π : M B is a Riemannian Submersion and M has positive sectional curvature, O Neill s Horizontal
More informationThe p-laplacian and geometric structure of the Riemannian manifold
Workshop on Partial Differential Equation and its Applications Institute for Mathematical Sciences - National University of Singapore The p-laplacian and geometric structure of the Riemannian manifold
More informationarxiv: v1 [math.dg] 25 Dec 2018 SANTIAGO R. SIMANCA
CANONICAL ISOMETRIC EMBEDDINGS OF PROJECTIVE SPACES INTO SPHERES arxiv:82.073v [math.dg] 25 Dec 208 SANTIAGO R. SIMANCA Abstract. We define inductively isometric embeddings of and P n (C) (with their canonical
More informationRIGIDITY OF STATIONARY BLACK HOLES WITH SMALL ANGULAR MOMENTUM ON THE HORIZON
RIGIDITY OF STATIONARY BLACK HOLES WITH SMALL ANGULAR MOMENTUM ON THE HORIZON S. ALEXAKIS, A. D. IONESCU, AND S. KLAINERMAN Abstract. We prove a black hole rigidity result for slowly rotating stationary
More informationtheorem for harmonic diffeomorphisms. Theorem. Let n be a complete manifold with Ricci 0, and let n be a simplyconnected manifold with nonpositive sec
on-existence of Some Quasi-conformal Harmonic Diffeomorphisms Lei i Λ Department of athematics University of California, Irvine Irvine, CA 92697 lni@math.uci.edu October 5 997 Introduction The property
More informationCOHOMOGENEITY ONE HYPERSURFACES OF EUCLIDEAN SPACES
COHOMOGENEITY ONE HYPERSURFACES OF EUCLIDEAN SPACES FRANCESCO MERCURI, FABIO PODESTÀ, JOSÉ A. P. SEIXAS AND RUY TOJEIRO Abstract. We study isometric immersions f : M n R n+1 into Euclidean space of dimension
More informationKUIPER S THEOREM ON CONFORMALLY FLAT MANIFOLDS
KUIPER S THEOREM ON CONFORMALLY FLAT MANIFOLDS RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA HOWARD@MATH.SC.EDU 1. Introduction These are notes to that show
More informationQing-Ming Cheng and Young Jin Suh
J. Korean Math. Soc. 43 (2006), No. 1, pp. 147 157 MAXIMAL SPACE-LIKE HYPERSURFACES IN H 4 1 ( 1) WITH ZERO GAUSS-KRONECKER CURVATURE Qing-Ming Cheng and Young Jin Suh Abstract. In this paper, we study
More informationarxiv: v1 [math.dg] 28 Aug 2017
SOE CLASSIFICATIONS OF BIHARONIC HYPERSURFACES WITH CONSTANT SCALAR CURVATURE SHUN AETA AND YE-LIN OU arxiv:708.08540v [math.dg] 28 Aug 207 Abstract We give some classifications of biharmonic hypersurfaces
More informationCOMPLETE SPACELIKE HYPERSURFACES IN THE DE SITTER SPACE
Chao, X. Osaka J. Math. 50 (203), 75 723 COMPLETE SPACELIKE HYPERSURFACES IN THE DE SITTER SPACE XIAOLI CHAO (Received August 8, 20, revised December 7, 20) Abstract In this paper, by modifying Cheng Yau
More informationStrictly convex functions on complete Finsler manifolds
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, No. 4, November 2016, pp. 623 627. DOI 10.1007/s12044-016-0307-2 Strictly convex functions on complete Finsler manifolds YOE ITOKAWA 1, KATSUHIRO SHIOHAMA
More informationSurfaces with Parallel Mean Curvature in S 3 R and H 3 R
Michigan Math. J. 6 (202), 75 729 Surfaces with Parallel Mean Curvature in S 3 R and H 3 R Dorel Fetcu & Harold Rosenberg. Introduction In 968, J. Simons discovered a fundamental formula for the Laplacian
More informationWilliam P. Thurston. The Geometry and Topology of Three-Manifolds
William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 00 http://www.msri.org/publications/books/gt3m/ This is an electronic edition of the 1980 notes distributed
More informationStability and Rigidity of Extremal Surfaces in Riemannian geometry and General Relativity
Surveys in Geometric Analysis and Relativity ALM 20, pp. 221 239 c Higher Education Press and International Press Beijing Boston Stability and Rigidity of Extremal Surfaces in Riemannian geometry and General
More informationMinimal submanifolds: old and new
Minimal submanifolds: old and new Richard Schoen Stanford University - Chen-Jung Hsu Lecture 1, Academia Sinica, ROC - December 2, 2013 Plan of Lecture Part 1: Volume, mean curvature, and minimal submanifolds
More informationHomework for Math , Spring 2012
Homework for Math 6170 1, Spring 2012 Andres Treibergs, Instructor April 24, 2012 Our main text this semester is Isaac Chavel, Riemannian Geometry: A Modern Introduction, 2nd. ed., Cambridge, 2006. Please
More informationRiemannian Curvature Functionals: Lecture III
Riemannian Curvature Functionals: Lecture III Jeff Viaclovsky Park City Mathematics Institute July 18, 2013 Lecture Outline Today we will discuss the following: Complete the local description of the moduli
More informationTHE GAUSS MAP OF TIMELIKE SURFACES IN R n Introduction
Chin. Ann. of Math. 16B: 3(1995),361-370. THE GAUSS MAP OF TIMELIKE SURFACES IN R n 1 Hong Jianqiao* Abstract Gauss maps of oriented timelike 2-surfaces in R1 n are characterized, and it is shown that
More informationarxiv: v1 [math.dg] 25 Nov 2009
arxiv:09.4830v [math.dg] 25 Nov 2009 EXTENSION OF REILLY FORULA WITH APPLICATIONS TO EIGENVALUE ESTIATES FOR DRIFTING LAPLACINS LI A, SHENG-HUA DU Abstract. In this paper, we extend the Reilly formula
More informationarxiv: v3 [math.dg] 13 Mar 2011
GENERALIZED QUASI EINSTEIN MANIFOLDS WITH HARMONIC WEYL TENSOR GIOVANNI CATINO arxiv:02.5405v3 [math.dg] 3 Mar 20 Abstract. In this paper we introduce the notion of generalized quasi Einstein manifold,
More informationCentre for Mathematics and Its Applications The Australian National University Canberra, ACT 0200 Australia. 1. Introduction
ON LOCALLY CONVEX HYPERSURFACES WITH BOUNDARY Neil S. Trudinger Xu-Jia Wang Centre for Mathematics and Its Applications The Australian National University Canberra, ACT 0200 Australia Abstract. In this
More informationVOLUME GROWTH AND HOLONOMY IN NONNEGATIVE CURVATURE
VOLUME GROWTH AND HOLONOMY IN NONNEGATIVE CURVATURE KRISTOPHER TAPP Abstract. The volume growth of an open manifold of nonnegative sectional curvature is proven to be bounded above by the difference between
More informationf -Minimal Surface and Manifold with Positive m-bakry Émery Ricci Curvature
J Geom Anal 2015) 25:421 435 DOI 10.1007/s12220-013-9434-5 f -inimal Surface and anifold with Positive m-bakry Émery Ricci Curvature Haizhong Li Yong Wei Received: 1 ovember 2012 / Published online: 16
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationA NOTE ON FISCHER-MARSDEN S CONJECTURE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 3, March 1997, Pages 901 905 S 0002-9939(97)03635-6 A NOTE ON FISCHER-MARSDEN S CONJECTURE YING SHEN (Communicated by Peter Li) Abstract.
More informationEuler Characteristic of Two-Dimensional Manifolds
Euler Characteristic of Two-Dimensional Manifolds M. Hafiz Khusyairi August 2008 In this work we will discuss an important notion from topology, namely Euler Characteristic and we will discuss several
More informationHYPERSURFACES OF EUCLIDEAN SPACE AS GRADIENT RICCI SOLITONS *
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul LXI, 2015, f.2 HYPERSURFACES OF EUCLIDEAN SPACE AS GRADIENT RICCI SOLITONS * BY HANA AL-SODAIS, HAILA ALODAN and SHARIEF
More informationStratification of 3 3 Matrices
Stratification of 3 3 Matrices Meesue Yoo & Clay Shonkwiler March 2, 2006 1 Warmup with 2 2 Matrices { Best matrices of rank 2} = O(2) S 3 ( 2) { Best matrices of rank 1} S 3 (1) 1.1 Viewing O(2) S 3 (
More informationMINIMIZING PROPERTIES OF CRITICAL POINTS OF QUASI-LOCAL ENERGY
MINIMIZING PROPERTIES OF CRITICAL POINTS OF QUASI-LOCAL ENERGY PO-NING CHEN, MU-TAO WANG, AND SHING-TUNG YAU arxiv:30.53v [math.dg] 5 Jun 03 Abstract. In relativity, the energy of a moving particle depends
More informationTobias Holck Colding: Publications. 1. T.H. Colding and W.P. Minicozzi II, Dynamics of closed singularities, preprint.
Tobias Holck Colding: Publications 1. T.H. Colding and W.P. Minicozzi II, Dynamics of closed singularities, preprint. 2. T.H. Colding and W.P. Minicozzi II, Analytical properties for degenerate equations,
More informationMath 6455 Nov 1, Differential Geometry I Fall 2006, Georgia Tech
Math 6455 Nov 1, 26 1 Differential Geometry I Fall 26, Georgia Tech Lecture Notes 14 Connections Suppose that we have a vector field X on a Riemannian manifold M. How can we measure how much X is changing
More informationStable minimal cones in R 8 and R 9 with constant scalar curvature
Revista Colombiana de Matemáticas Volumen 6 (2002), páginas 97 106 Stable minimal cones in R 8 and R 9 with constant scalar curvature Oscar Perdomo* Universidad del Valle, Cali, COLOMBIA Abstract. In this
More informationHADAMARD FOLIATIONS OF H n. I
HADAMARD FOLIATIONS OF H n. I MACIEJ CZARNECKI Abstract. We introduce the notion of an Hadamard foliation as a foliation of Hadamard manifold which all leaves are Hadamard. We prove that a foliation of
More informationthe neumann-cheeger constant of the jungle gym
the neumann-cheeger constant of the jungle gym Itai Benjamini Isaac Chavel Edgar A. Feldman Our jungle gyms are dimensional differentiable manifolds M, with preferred Riemannian metrics, associated to
More informationDeforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary
Deforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary Weimin Sheng (Joint with Li-Xia Yuan) Zhejiang University IMS, NUS, 8-12 Dec 2014 1 / 50 Outline 1 Prescribing
More informationDIFFERENTIAL GEOMETRY HW 12
DIFFERENTIAL GEOMETRY HW 1 CLAY SHONKWILER 3 Find the Lie algebra so(n) of the special orthogonal group SO(n), and the explicit formula for the Lie bracket there. Proof. Since SO(n) is a subgroup of GL(n),
More informationThe Geometrization Theorem
The Geometrization Theorem Matthew D. Brown Wednesday, December 19, 2012 In this paper, we discuss the Geometrization Theorem, formerly Thurston s Geometrization Conjecture, which is essentially the statement
More informationNegative sectional curvature and the product complex structure. Harish Sheshadri. Department of Mathematics Indian Institute of Science Bangalore
Negative sectional curvature and the product complex structure Harish Sheshadri Department of Mathematics Indian Institute of Science Bangalore Technical Report No. 2006/4 March 24, 2006 M ath. Res. Lett.
More informationQuasi-local Mass and Momentum in General Relativity
Quasi-local Mass and Momentum in General Relativity Shing-Tung Yau Harvard University Stephen Hawking s 70th Birthday University of Cambridge, Jan. 7, 2012 I met Stephen Hawking first time in 1978 when
More informationA SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY
A SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY PLAMEN STEFANOV 1. Introduction Let (M, g) be a compact Riemannian manifold with boundary. The geodesic ray transform I of symmetric 2-tensor fields f is
More informationGEOMETRY OF GEODESIC SPHERES IN A COMPLEX PROJECTIVE SPACE IN TERMS OF THEIR GEODESICS
Mem. Gra. Sci. Eng. Shimane Univ. Series B: Mathematics 51 (2018), pp. 1 5 GEOMETRY OF GEODESIC SPHERES IN A COMPLEX PROJECTIVE SPACE IN TERMS OF THEIR GEODESICS SADAHIRO MAEDA Communicated by Toshihiro
More informationON HAMILTONIAN STATIONARY LAGRANGIAN SPHERES IN NON-EINSTEIN KÄHLER SURFACES
ON HAMILTONIAN STATIONARY LAGRANGIAN SPHERES IN NON-EINSTEIN KÄHLER SURFACES ILDEFONSO CASTRO, FRANCISCO TORRALBO, AND FRANCISCO URBANO Abstract. Hamiltonian stationary Lagrangian spheres in Kähler-Einstein
More informationHILBERT S THEOREM ON THE HYPERBOLIC PLANE
HILBET S THEOEM ON THE HYPEBOLIC PLANE MATTHEW D. BOWN Abstract. We recount a proof of Hilbert s result that a complete geometric surface of constant negative Gaussian curvature cannot be isometrically
More informationarxiv:math/ v1 [math.dg] 19 Nov 2004
arxiv:math/04426v [math.dg] 9 Nov 2004 REMARKS ON GRADIENT RICCI SOLITONS LI MA Abstract. In this paper, we study the gradient Ricci soliton equation on a complete Riemannian manifold. We show that under
More informationMinimization problems on the Hardy-Sobolev inequality
manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev
More informationSOME ASPECTS ON CIRCLES AND HELICES IN A COMPLEX PROJECTIVE SPACE. Toshiaki Adachi* and Sadahiro Maeda
Mem. Fac. Sci. Eng. Shimane Univ. Series B: Mathematical Science 32 (1999), pp. 1 8 SOME ASPECTS ON CIRCLES AND HELICES IN A COMPLEX PROJECTIVE SPACE Toshiaki Adachi* and Sadahiro Maeda (Received December
More informationChapter 16. Manifolds and Geodesics Manifold Theory. Reading: Osserman [7] Pg , 55, 63-65, Do Carmo [2] Pg ,
Chapter 16 Manifolds and Geodesics Reading: Osserman [7] Pg. 43-52, 55, 63-65, Do Carmo [2] Pg. 238-247, 325-335. 16.1 Manifold Theory Let us recall the definition of differentiable manifolds Definition
More informationMinimal hypersurfaces with bounded index and area
Minimal hypersurfaces with bounded index and area Ben Sharp March 2017 Why should one care? Set-up and definitions Compactness and non-compactness Main results Why should one care? The work of Almgren
More informationarxiv: v4 [math.dg] 18 Jun 2015
SMOOTHING 3-DIMENSIONAL POLYHEDRAL SPACES NINA LEBEDEVA, VLADIMIR MATVEEV, ANTON PETRUNIN, AND VSEVOLOD SHEVCHISHIN arxiv:1411.0307v4 [math.dg] 18 Jun 2015 Abstract. We show that 3-dimensional polyhedral
More informationUmbilic cylinders in General Relativity or the very weird path of trapped photons
Umbilic cylinders in General Relativity or the very weird path of trapped photons Carla Cederbaum Universität Tübingen European Women in Mathematics @ Schloss Rauischholzhausen 2015 Carla Cederbaum (Tübingen)
More informationMetric and comparison geometry
Surveys in Differential Geometry XI Metric and comparison geometry Jeff Cheeger and Karsten Grove The present volume surveys some of the important recent developments in metric geometry and comparison
More informationHARMONIC FUNCTIONS, ENTROPY, AND A CHARACTERIZATION OF THE HYPERBOLIC SPACE
HARMONIC FUNCTIONS, ENTROPY, AND A CHARACTERIATION OF THE HYPERBOLIC SPACE XIAODONG WANG Abstract. Let (M n ; g) be a compact Riemannian manifold with Ric (n ). It is well known that the bottom of spectrum
More informationAsymptotic Behavior of Marginally Trapped Tubes
Asymptotic Behavior of Marginally Trapped Tubes Catherine Williams January 29, 2009 Preliminaries general relativity General relativity says that spacetime is described by a Lorentzian 4-manifold (M, g)
More informationarxiv: v1 [math.dg] 8 Jan 2019
THE OBATA EQUATION WITH ROBIN BOUNDARY CONDITION XUEZHANG CHEN, MIJIA LAI, AND FANG WANG arxiv:1901.02206v1 [math.dg] 8 Jan 2019 Abstract. We study the Obata equation with Robin boundary condition f +
More informationHarmonic measures on negatively curved manifolds
Harmonic measures on negatively curved manifolds Yves Benoist & Dominique Hulin Abstract We prove that the harmonic measures on the spheres of a pinched Hadamard manifold admit uniform upper and lower
More informationOLIVIA MILOJ March 27, 2006 ON THE PENROSE INEQUALITY
OLIVIA MILOJ March 27, 2006 ON THE PENROSE INEQUALITY Abstract Penrose presented back in 1973 an argument that any part of the spacetime which contains black holes with event horizons of area A has total
More informationarxiv: v2 [math.dg] 25 Oct 2014
GAP PHENOMENA AND CURVATURE ESTIMATES FOR CONFORMALLY COMPACT EINSTEIN MANIFOLDS GANG LI, JIE QING AND YUGUANG SHI arxiv:1410.6402v2 [math.dg] 25 Oct 2014 Abstract. In this paper we first use the result
More informationComplete Surfaces of Constant Gaussian Curvature in Euclidean Space R 3.
Summary of the Thesis in Mathematics by Valentina Monaco Complete Surfaces of Constant Gaussian Curvature in Euclidean Space R 3. Thesis Supervisor Prof. Massimiliano Pontecorvo 19 May, 2011 SUMMARY The
More informationUNIQUENESS OF STATIC BLACK-HOLES WITHOUT ANALYTICITY. Piotr T. Chruściel & Gregory J. Galloway
UNIQUENESS OF STATIC BLACK-HOLES WITHOUT ANALYTICITY by Piotr T. Chruściel & Gregory J. Galloway Abstract. We show that the hypothesis of analyticity in the uniqueness theory of vacuum, or electrovacuum,
More informationTobias Holck Colding: Publications
Tobias Holck Colding: Publications [1] T.H. Colding and W.P. Minicozzi II, The singular set of mean curvature flow with generic singularities, submitted 2014. [2] T.H. Colding and W.P. Minicozzi II, Lojasiewicz
More information1 First and second variational formulas for area
1 First and second variational formulas for area In this chapter, we will derive the first and second variational formulas for the area of a submanifold. This will be useful in our later discussion on
More informationLocalizing solutions of the Einstein equations
Localizing solutions of the Einstein equations Richard Schoen UC, Irvine and Stanford University - General Relativity: A Celebration of the 100th Anniversary, IHP - November 20, 2015 Plan of Lecture The
More informationESTIMATES ON THE GRAPHING RADIUS OF SUBMANIFOLDS AND THE INRADIUS OF DOMAINS
ESTIMATES ON THE GRAPHING RADIUS OF SUBMANIFOLDS AND THE INRADIUS OF DOMAINS RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA HOWARD@MATH.SC.EDU Abstract. Let
More informationarxiv:gr-qc/ v1 24 Feb 2004
A poor man s positive energy theorem arxiv:gr-qc/0402106v1 24 Feb 2004 Piotr T. Chruściel Département de Mathématiques Faculté des Sciences Parc de Grandmont F37200 Tours, France Gregory J. Galloway Department
More informationSharp gradient estimate and spectral rigidity for p-laplacian
Shar gradient estimate and sectral rigidity for -Lalacian Chiung-Jue Anna Sung and Jiaing Wang To aear in ath. Research Letters. Abstract We derive a shar gradient estimate for ositive eigenfunctions of
More informationLECTURE 10: THE PARALLEL TRANSPORT
LECTURE 10: THE PARALLEL TRANSPORT 1. The parallel transport We shall start with the geometric meaning of linear connections. Suppose M is a smooth manifold with a linear connection. Let γ : [a, b] M be
More informationH-convex Riemannian submanifolds
H-convex Riemannian submanifolds Constantin Udrişte and Teodor Oprea Abstract. Having in mind the well known model of Euclidean convex hypersurfaces [4], [5] and the ideas in [1], many authors defined
More informationPseudo-Poincaré Inequalities and Applications to Sobolev Inequalities
Pseudo-Poincaré Inequalities and Applications to Sobolev Inequalities Laurent Saloff-Coste Abstract Most smoothing procedures are via averaging. Pseudo-Poincaré inequalities give a basic L p -norm control
More informationTHE GEOMETRY OF OPEN MANIFOLDS OF NONNEGATIVE CURVATURE. Ph.D. Thesis, University of Pennsylvania, Kristopher R. Tapp
THE GEOMETRY OF OPEN MANIFOLDS OF NONNEGATIVE CURVATURE Ph.D. Thesis, University of Pennsylvania, 1999 Kristopher R. Tapp Acknowledgments I wish to thank Wolfgang Ziller for showing me the beauty of Differential
More informationA New Proof of Lee's Theorem on the Spectrum of Conformally Compact Einstein Manifolds
COMMUNICATIONS IN ANALYSIS AND GEOMETRY Volume 10, Number 3, 647-651, 2002 A New Proof of Lee's Theorem on the Spectrum of Conformally Compact Einstein Manifolds XIAODONG WANG Let M be a compact n + 1-dimensional
More informationThe existence of light-like homogeneous geodesics in homogeneous Lorentzian manifolds. Sao Paulo, 2013
The existence of light-like homogeneous geodesics in homogeneous Lorentzian manifolds Zdeněk Dušek Sao Paulo, 2013 Motivation In a previous project, it was proved that any homogeneous affine manifold (and
More informationDIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17
DIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17 6. Geodesics A parametrized line γ : [a, b] R n in R n is straight (and the parametrization is uniform) if the vector γ (t) does not depend on t. Thus,
More information