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.     

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.     

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.     

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.     

Symplectic duality of symmetric spaces

Let M⊂Cn be a complex n-dimensional Hermitian symmetric space endowed with the hyperbolic form ωhyp. Denote by (M^∗,ωFS) the compact dual of (M,ωhyp), where ωFS is the Fubini-Study form on M^∗. Our first result is Theorem 1.1 where, with the aid of the theory of Jordan triple systems, we construct an explicit symplectic duality, namely a diffeomorphism satisfying and for the pull-back of ΨM, where ω₀ is the restriction to M of the flat Kähler form of the Hermitian positive Jordan triple system associated to M. Amongst other properties of the map ΨM, we also show that it takes (complete) complex and totally geodesic submanifolds of M through the origin to complex linear subspaces of Cn. As a byproduct of the proof of Theorem 1.1 we get an interesting characterization (Theorem 5.3) of the Bergman form of a Hermitian symmetric space in terms of its restriction to classical complex and totally geodesic submanifolds passing through the origin.     

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 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.     

Positive sectional curvature, symmetry and Chern's conjecture

Let M be a n-manifold of positive sectional curvature. Suppose that M admits an isometrical torus Tk-action with . The main results of the paper are: (1) the fundamental group π₁(M) contains no Zp⊕Zp subgroup with p prime and p≠3 (a partial positive answer to Chern's conjecture); (2) the 2-order element of π₁(M) belongs to the center of π₁(M).     

Liouville type theorems for p-harmonic maps

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that at all x∈M and at some point x₀∈M, where μ₀>0 is the least eigenvalue of the Laplacian acting on L²-functions on M. Let 2?q?p. Then any q-harmonic map of finite q-energy is constant. Moreover, if N is a Riemannian manifold of non-positive scalar curvature, then any q-harmonic morphism of finite q-energy is constant.     

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].     

Sharp estimates for the non-centered maximal operator associated to Gaussian and other radial measures

In this work we investigate the integrability properties of the maximal operator M_μ, associated with a non-doubling measure μ defined on the Euclidean space , with special emphasis on the Gaussian and similar measures. Among other results we show for a wide class of radial and decreasing measures μ, that M_μ satisfies the modular inequality

     

4-dimensional minimal CR submanifolds of the sphere S satisfying Chen's equality

In this paper, we classify 4-dimensional minimal CR submanifolds M of the nearly Kähler 6-sphere S⁶(1) which satisfy Chen's equality, i.e. , where δM(p)=τ(p)−infK(p) for every p∈M.     

A uniform bound for the solutions to a simple nonlinear equation on Riemannian manifolds

Let M be a complete noncompact manifold with Ricci curvature bounded below. In this note, we derive a uniform bound for the solutions to the nonlinear equation

     

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.     

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.