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.     

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

In this paper, we classify complete spacelike hypersurfaces in the anti-de Sitter space (n?3) 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 also obtain several rigidity theorems for such hypersurfaces.     

An extension of Jellett's theorem

The aim of this work is to show that a star-shaped hypersurface of constant mean curvature into the Euclidean sphere Sn+1 must be a geodesic sphere. This result extends the one obtained by Jellett in 1853 for such type of surfaces in the Euclidean space R³. In order to do that we will compute a useful formula for the Laplacian of a new support function defined over a hypersurface M of a Riemannian manifold .     

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.     

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.     

Uniform stabilization of the wave equation on compact manifolds and locally distributed damping - a sharp result

Let (M,g) be an n-dimensional (n?2) compact Riemannian manifold with or without boundary where g denotes a Riemannian metric of class C^∞. This paper is concerned with the study of the wave equation on (M,g) with locally distributed damping, described by

     

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.     

On the Ricci curvature of steady gradient Ricci solitons

Assume (Mn,g) is a complete steady gradient Ricci soliton with positive Ricci curvature. If the scalar curvature approaches 0 towards infinity, we prove that , where O is the point where R obtains its maximum and γ(s) is a minimal normal geodesic emanating from O. Some other results on the Ricci curvature are also obtained.     

Collapse of warped foliations

The aim of this note is to generalize the concept of warped product to a foliated manifold (M,F,g) as follows: If is a smooth function constant along the leaves of the foliation F then new metric structure gf on the manifold M is constructed as follows: gf(v,w)=f²g(v,w) if v,w are tangent to F and gf(v,w)=g(v,w) if v or w is perpendicular to F. A foliated manifold (M,F,gf) is called warped foliation while f is called warping function.Next, if is a sequence of warping functions on M, the question of the existence of the limit in Gromov-Hausdorff of a sequence ((M,F,gfn))_n∈N warped foliation is asked. A number of examples is considered such foliations with dense leaf or foliations consisting of finite number of Reeb components. Next, sufficient and necessary condition of converging in Gromov-Hausdorff sense of a Riemannian foliation with all leaves compact to the space of leaves with a metric defined by Hausdorff distance of leaves is developed. Finally some results on Hausdorff foliations with all leaves compact are shown.     

Pinching theorems of hypersurfaces in a unit sphere

Let Mn be a complete hypersurface in Sn+1(1) with constant mean curvature. Assume that Mn has n−1 principal curvatures with the same sign everywhere. We prove that if RicM≤C₋(H), either S?S₊(H) or RicM?0 or the fundamental group of Mn is infinite, then S is constant, S=S₊(H) and Mn is isometric to a Clifford torus with . These rigidity theorems are still valid for compact hypersurface without constancy condition on the mean curvature.     

On a class of twistorial maps

We show that a natural class of twistorial maps gives a pattern for apparently different geometric maps, such as, (1,1)-geodesic immersions from (1,2)-symplectic almost Hermitian manifolds and pseudo horizontally conformal submersions with totally geodesic fibres for which the associated almost CR-structure is integrable. Along the way, we construct for each constant curvature Riemannian manifold (M,g), of dimension m, a family of twistor spaces such that Zr(M) parametrizes naturally the set of pairs (P,J), where P is a totally geodesic submanifold of (M,g), of codimension 2r, and J is an orthogonal complex structure on the normal bundle of P which is parallel with respect to the normal connection.     

The Hopf conjecture for positively curved manifolds with discrete abelian group actions

Let M be a closed even n-manifold of positive sectional curvature. The main result asserts that the Euler characteristic of M is positive, if M admits an isometric -action with prime p?p(n) (a constant depending only on n) and k satisfies any one of the following conditions: (i) and n≠12, 18 or 20; (ii) , and n≡0 mod 4 with n≠12 or 20; (iii) , and n≡0,4 or 12 mod 20 with n≠20. This generalizes some results in [T. Püttmann, C. Searle, The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank, Proc. Amer. Math. Soc. 130 (2002) 163-166; X. Rong, Positively curved manifolds with almost maximal symmetry rank, Geom. Dedicata 59 (2002) 157-182; X. Rong, X. Su, The Hopf conjecture for positively curved manifolds with abelian group actions, Comm. Cont. Math. 7 (2005) 121-136].     

Locally strongly convex hypersurfaces with constant affine mean curvature

Let be a locally strongly convex hypersurface, given by a strictly convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂An. We consider the Riemannian metric G# on M, defined by . In this paper we prove that if M is a locally strongly convex surface with constant affine mean curvature and if M is complete with respect to the metric G#, then M must be an elliptic paraboloid.     

Eigenvalue estimate of the basic Dirac operator on a Kähler foliation

Let F be a Kähler spin foliation of codimension q=2n on a compact Riemannian manifold M with the transversally holomorphic mean curvature form κ. It is well known [S.D. Jung, T.H. Kang, Lower bounds for the eigenvalue of the transversal Dirac operator on a Kähler foliation, J. Geom. Phys. 45 (2003) 75-90] that the eigenvalue λ of the basic Dirac operator Db satisfies the inequality , where σ∇ is the transversal scalar curvature of F. In this paper, we introduce the transversal Kählerian twistor operator and prove that the same inequality for the eigenvalue of the basic Dirac operator by using the transversal Kählerian twistor operator. We also study the limiting case. In fact, F is minimal and transversally Einsteinian of odd complex codimension n with nonnegative constant transversal scalar curvature.     

Heteroclinic solutions for a class of the second order Hamiltonian systems

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system , where q∈Rn and V∈C¹(R×Rn,R), V?0. We will assume that V and a certain subset M⊂Rn satisfy the following conditions. M is a set of isolated points and #M?2. For every sufficiently small ε>0 there exists δ>0 such that for all (t,z)∈R×Rn, if d(z,M)?ε then −V(t,z)?δ. The integrals , z∈M, are equi-bounded and −V(t,z)→∞, as |t|→∞, uniformly on compact subsets of Rn?M. Our result states that each point in M is joined to another point in M by a solution of our system.     

Interior estimates for solutions of a fourth order nonlinear partial differential equation

Let be a locally strongly convex hypersurface, given by the graph of a convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂Rn. M is called a α-extremal hypersurface, if f is a solution of

     

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?     

Smooth approximation of Lipschitz functions on Riemannian manifolds

We show that for every Lipschitz function f defined on a separable Riemannian manifold M (possibly of infinite dimension), for every continuous , and for every positive number r>0, there exists a C^∞ smooth Lipschitz function such that |f(p)−g(p)|?ε(p) for every p∈M and Lip(g)?Lip(f)+r. Consequently, every separable Riemannian manifold is uniformly bumpable. We also present some applications of this result, such as a general version for separable Riemannian manifolds of Deville-Godefroy-Zizler's smooth variational principle.     

Linearized inverse problem for the Dirichlet-to-Neumann map on differential forms

For a compact n-dimensional Riemannian manifold (M,g) with boundary i:∂M⊂M, the Dirichlet-to-Neumann (DN) map Λg:Ωk(∂M)→Ωn−k−1(∂M) is defined on exterior differential forms by Λgφ=i^∗(?dω), where ω solves the boundary value problem Δω=0, i^∗ω=φ, i^∗δω=0. For a symmetric second rank tensor field h on M, let be the Gateaux derivative of the DN map in the direction h. We study the question: for a given (M,g), how large is the subspace of tensor fields h satisfying ? Potential tensor fields belong to the subspace since the DN map is invariant under isomeries fixing the boundary. For a manifold of an even dimension n, the DN map on (n/2−1)-forms is conformally invariant, therefore spherical tensor fields belong to the subspace in the case of k=n/2−1. The manifold is said to be Ωk-rigid if there is no other h satisfying . We prove that the Ωk-rigidity is equivalent to the density of the range of some bilinear form on the space of exact harmonic fields.     

The mapping properties of filling radius and packing radius and their applications

Let be a map between closed, oriented Riemannian n-manifolds. It is shown that FillRad(W)?dil(f)⋅FillRad(V), if |deg(f)|=1. By this mapping property, we obtain an estimate from below for the filling radius of a closed, oriented, nonpositively curved manifold, or a manifold with sectional curvature bounded above by a positive constant. In addition, a similar mapping property of packing radius and a corollary are also obtained.