Stability of spacelike hypersurfaces in foliated spacetimes

Given a generalized Robertson-Walker spacetime whose warping function verifies a certain convexity condition, we classify strongly stable spacelike hypersurfaces with constant mean curvature. More precisely, we will show that given a closed, strongly stable spacelike hypersurface of with constant mean curvature H, if the warping function ? satisfying ?^″?max{H?^′,0} along M, then Mn is either maximal or a spacelike slice Mt₀={t₀}×F, for some t₀∈I.     

Extrinsic radius pinching for hypersurfaces of space forms

We prove some pinching results for the extrinsic radius of compact hypersurfaces in space forms. In the hyperbolic space, we show that if the volume of M is 1, then there exists a constant C depending on the dimension of M and the L^∞-norm of the second fundamental form B such that the pinching condition (where H is the mean curvature) implies that M is diffeomorphic to an n-dimensional sphere. We prove the corresponding result for hypersurfaces of the Euclidean space and the sphere with the Lp-norm of H, p?2, instead of the L^∞-norm.     

Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in De Sitter space

In this paper we give a partially affirmative answer to the following question posed by Haizhong Li: is a complete spacelike hypersurface in De Sitter space , n?3, with constant normalized scalar curvature R satisfying totally umbilical?     

A new variational characterization of the Wulff shape

Given a positive function F on Sn which satisfies a convexity condition, we define the rth anisotropic mean curvature function Mr for hypersurfaces in Rn+1 which is a generalization of the usual rth mean curvature function. Let be an n-dimensional closed hypersurface with , for some r with 1?r?n−1, which is a critical point for a variational problem. We show that X(M) is stable if and only if X(M) is the Wulff shape.     

Flot de courbure moyenne modifiée avec obstacle conique

A rotationally symmetric, compact, oriented, connected, uniformly convex hypersurface M₀ of , with boundary ∂M₀ in a rotationally symmetric cone S, is evolving under volume-preserving mean curvature flow. Then for n?2, we obtain gradient and curvature estimates, leading to long-time existence of the flow, and convergence to a part of a round sphere.     

δ-invariants and their applications to centroaffine geometry

We introduce the notion of δ-invariant for curvature-like tensor fields and establish optimal general inequalities in case the curvature-like tensor field satisfies some algebraic Gauss equation. We then study the situation when the equality case of one of the inequalities is satisfied and prove a dimension and decomposition theorem. In the second part of the paper, we apply these results to definite centroaffine hypersurfaces in Rn+1. The inequality is specified into an inequality involving the affine δ-invariants and the Tchebychev vector field. We show that if a centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is a proper affine hypersphere. Furthermore, we prove that if a positive definite centroaffine hypersurface in , satisfies the equality case of one of the inequalities, it is foliated by ellipsoids. And if a negative definite centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is foliated by two-sheeted hyperboloids. Some further applications of the inequalities are also provided in this article.     

On spacelike hypersurfaces with constant scalar curvature in the de Sitter space

We classify spacelike hypersurfaces of the de Sitter space with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we prove several rigidity theorems for such hypersurfaces.     

Characterizing the round sphere by mean distance

We discuss the measure theoretic metric invariants extent, rendezvous number and mean distance of a general compact metric space X and relate these to classical metric invariants such as diameter and radius. In the final section we focus attention to the category of Riemannian manifolds. The main result of this paper is Theorem 4 stating that the round sphere of constant curvature 1 has maximal mean distance among Riemannian n-manifolds with Ricci curvature Ric?n−1, and that such a manifold is diffeomorphic to a sphere if the mean distance is close to .     

U(1)-invariant special Lagrangian 3-folds. III. Properties of singular solutions

This is the last of three papers studying special Lagrangian 3-submanifolds (SLV 3-folds) N in invariant under the U(1)-action e^iθ:(z₁,z₂,z₃)?(e^iθz₁,e^iθz₂,z₃), using analytic methods. If N is such a 3-fold then |z₁|²−|z₂|²=2a on N for some . Locally, N can be written as a kind of graph of functions satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy.The first paper studied the case a nonzero, and proved existence and uniqueness for solutions of two Dirichlet problems derived from the nonlinear Cauchy-Riemann equation. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in , with boundary conditions. The second paper extended these results to weak solutions of the Dirichlet problems when a=0, giving existence and uniqueness of many singular U(1)-invariant SL 3-folds in , with boundary conditions.This third paper studies the singularities of these SL 3-folds. We show that under mild conditions the singularities are isolated, and have a multiplicityn>0, and one of two types. Examples are constructed with every multiplicity and type. We also prove the existence of large families of U(1)-invariant special Lagrangian fibrations of open sets in , including singular fibres.     

On the Ricci and Einstein equations on the pseudo-euclidean and hyperbolic spaces

We consider tensors T=fg on the pseudo-euclidean space Rn and on the hyperbolic space Hn, where n?3, g is the standard metric and f is a differentiable function. For such tensors, we consider, in both spaces, the problems of existence of a Riemannian metric , conformal to g, such that , and the existence of such a metric which satisfies , where is the scalar curvature of . We find the restrictions on the Ricci candidate for solvability and we construct the solutions when they exist. We show that these metrics are unique up to homothety, we characterize those globally defined and we determine the singularities for those which are not globally defined. None of the non-homothetic metrics , defined on Rn or Hn, are complete. As a consequence of these results, we get positive solutions for the equation , where g is the pseudo-euclidean metric.     

Harmonicity of unit vector fields with respect to Riemannian g-natural metrics

Let (M,g) be a compact Riemannian manifold and T₁M its unit tangent sphere bundle. Unit vector fields defining harmonic maps from (M,g) to , being the Sasaki metric on T₁M, have been extensively studied. The Sasaki metric, and other well known Riemannian metrics on T₁M, are particular examples of g-natural metrics. We equip T₁M with an arbitrary Riemannian g-natural metric , and investigate the harmonicity of a unit vector field V of M, thought as a map from (M,g) to . We then apply this study to characterize unit Killing vector fields and to investigate harmonicity properties of the Reeb vector field of a contact metric manifold.     

On the twistor space of pseudo-spheres

We give a proof that the sphere S⁶ does not admit an integrable orthogonal complex structure using simple differential geometric methods. This appears as a corollary of a general analogous result concerning pseudo-spheres.We study the twistor space of a pseudo-Riemannian manifold in both the holomorphic and pseudo-Riemannian directions. In particular, we construct the twistor space of a pseudo-sphere as a known pseudo-Kähler symmetric space. This leads to the explicit, unexpected computation of the exterior derivative of the Kähler form on the base manifold.     

U(1)-invariant special Lagrangian 3-folds. I. Nonsingular solutions

This is the first of three papers studying special Lagrangian 3-submanifolds (SL 3-folds) N in invariant under the U(1)-action e^iθ:(z₁,z₂,z₃)?(e^iθz₁,e^−iθz₂,z₃), using analytic methods.Let N be such a U(1)-invariant SL 3-fold. Then |z₁|²−|z₂|²=2a on N for some . Locally, N can be written as a kind of graph of functions satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy. When a is nonzero, u,v are always smooth and N is always nonsingular. But if a=0, there may be points (x,0) where u,v are not differentiable, which correspond to singular points of N.This paper focusses on the nonsingular case, when a is nonzero. We prove analogues for our nonlinear Cauchy-Riemann equation of well-known results in complex analysis. In particular, we prove existence and uniqueness for solutions of two Dirichlet problems derived from it. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in , with two kinds of boundary conditions. The sequels extend these to the case a=0, study the singularities of the SL 3-folds that arise, and construct special Lagrangian fibrations of open sets in .     

Isometric embeddings of real projective spaces into Euclidean spaces

This paper studies isometric embeddings of RPn via non-degenerate symmetric bilinear maps. The main result shows the infimum dimension of target Euclidean spaces among these constructions for RPn is . Next, we construct Veronese maps by induction, which realize the infimum. Finally, we give a simple proof of Rigidity Theorem of Veronese maps.     

U(1)-invariant special Lagrangian 3-folds. II. Existence of singular solutions

This is the second of three papers studying special Lagrangian 3-submanifolds (SLV 3-folds) N in invariant under the U(1)-action e^iθ:(z₁,z₂,z₃)?(e^iθz₁,e^−iθz₂,z₃), using analytic methods. If N is such a 3-fold then |z₁|²−|z₂|²=2a on N for some . Locally, N can be written as a kind of graph of functions satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy.The first paper studied the case when a is nonzero. Then u,v are smooth and N is nonsingular. It proved existence and uniqueness for solutions of two Dirichlet problems derived from the equations on u,v. This implied existence and uniqueness for a large class of nonsingular U(1)-invariant SL 3-folds in , with boundary conditions.In this paper and its sequel we focus on the case a=0. Then the nonlinear Cauchy-Riemann equation is not always elliptic. Because of this there may be points (x,0) where u,v are not differentiable, corresponding to singular points of N. This paper is concerned largely with technical analytic issues, and the sequel with the geometry of the singularities of N. We prove a priori estimates for derivatives of solutions of the nonlinear Cauchy-Riemann equation, and use them to show existence and uniqueness of weak solutions u,v to the two Dirichlet problems when a=0, which are continuous and weakly differentiable. This gives existence and uniqueness for a large class of singular U(1)-invariant SL 3-folds in , with boundary conditions.     

Classification of pinched positive scalar curvature manifolds

Let (Mn,g), n?3, be a smooth closed Riemannian manifold with positive scalar curvature Rg. There exists a positive constant C=C(M,g) defined by mean curvature of Euclidean isometric immersions, which is a geometric invariant, such that Rg?n(n−1)C. In this paper we prove that Rg=n(n−1)C if and only if (Mn,g) is isometric to the Euclidean sphere Sn(C) with constant sectional curvature C. Also, there exists a Riemannian metric g on Mn such that the scalar curvature satisfies the pinched condition