Height estimates and half-space type theorems in weighted product spaces with nonnegative Bakry–Émery–Ricci curvature

We prove height estimates concerning compact hypersurfaces with nonzero constant weighted mean curvature and whose boundary is contained into a slice of a weighted product space of nonnegative Bakry–Émery–Ricci curvature. As applications of our estimates, we obtain half-space type results related to complete noncompact hypersurfaces properly immersed in such an ambient space.     

Essential spectrum of the weighted Laplacian on noncompact manifolds and applications

We obtain upper estimates for the bottom (that is, greatest lower bound) of the essential spectrum of weighted Laplacian operator of a noncompact weighted manifold under assumptions of the volume growth of their geodesic balls and spheres. Furthermore, we find examples where the equality occurs in the estimates obtained. As a consequence, we give estimates for the weighted mean curvature of complete noncompact hypersurfaces into weighted manifolds.     

Rigidity theorems for complete $$\lambda $$ λ -hypersurfaces

Archiv der Mathematik - In this article, we study hypersurfaces $$\Sigma \subset {\mathbb {R}}^{n+1}$$ with constant weighted mean curvature, also known as $$\lambda $$ -hypersurfaces. Recently,...     

Bifurcations of immersed constant mean curvature hypersurfaces in hyperbolic space

In this paper, we prove the existence of new branches of hypersurfaces with constant mean curvature which bifurcate from the rotationally invariant immersed constant mean curvature hypersurfaces in the hyperbolic space.     

Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions

Inspired by earlier results on the quasilinear mean curvature flow, and recent investigations of fully nonlinear curvature flow of closed hypersurfaces which are not convex, we consider contraction of axially symmetric hypersurfaces by convex, degree-one homogeneous fully nonlinear functions of curvature. With a natural class of Neumann boundary conditions, we show that evolving hypersurfaces exist for a finite maximal time. The maximal time is characterised by a curvature singularity at either boundary. Some results continue to hold in the cases of mixed Neumann–Dirichlet boundary conditions and more general curvature-dependent speeds.     

A NOTE ON THE MEAN CURVATURE FLOW IN RIEMANNIAN MANIFOLDS

Under the hypothesis of mean curvature flows of hypersurfaces, we prove that the limit of the smooth rescaling of the singularity is weakly convex. It is a generalization of the result due to G.Huisken and C. Sinestrari in. These apriori bounds are satisfied for mean convex hypersurfaces in locally symmetric Riemannian manifolds with nonnegative sectional curvature.     

Gluing constructions amongst constant mean curvature hypersurfaces in $${\mathbb {S}^{n+1}}$$

Four constructions of constant mean curvature (CMC) hypersurfaces in \mathbb Sⁿ⁺¹{\mathbb {S}^{n+1}} are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all the values of the mean curvature. Second, a hypersurface is constructed which consists of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused together in a symmetric manner. Third, a Delaunay-like handle can be attached to a generalized Clifford torus of the same mean curvature. Finally, two generalized Clifford tori of equal but opposite mean curvature of any magnitude can be attached to each other by symmetrically positioned Delaunay-like ‘arms’. This last result extends Butscher and Pacard’s doubling construction for generalized Clifford tori of small mean curvature.     

Lorentz-Minkowski空间中的乘积空间R~k×H~(n-k)

本文的主要结果是:Lorentz-Minkowski空间中介于两个同心伪圆柱面之间的完备、类空、常平均曲率超曲面必为伪圆柱面,即乘积空间R~k×H~(n-k)．对于常高阶平均曲率的情况,如果截曲率有下界,那么介于两个同心伪球面之间的完备类空超曲面必为伪球面．     

S4中等参超曲面的谱刻画

本文证明了如果Ｓ⁴中的具常平均曲率ｈ的超曲面Ｍ与其具平均曲率ｈ的等参超曲面Ｍ_０（强）等谱，则Ｍ＝Ｍ_０．     

Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces

In this paper we give pinching theorems for the first nonzero eigenvalue of the Laplacian on compact hypersurfaces of ambient spaces with bounded sectional curvature. As an application we deduce a rigidity result for stable constant mean curvature hypersurfaces M of these spaces N. Indeed, we prove that if M is included in a ball of radius small enough then the Hausdorff-distance between M and a geodesic sphere S of N is small. Moreover M is diffeomorphic and quasi-isometric to S. As other application, we obtain rigidity results for almost umbilic hypersurfaces.     

On the geometry of conformally stationary Lorentz spaces

We study several aspects of the geometry of conformally stationary Lorentz manifolds, and particularly of GRW spaces, due to the presence of a closed conformal vector field. More precisely, we begin by extending a result of J. Simons on the minimality of cones in Euclidean space to these spaces, and apply it to the construction of complete, noncompact minimal Lorentz submanifolds of both de Sitter and anti-de Sitter spaces. Then we state and prove very general Bernstein-type theorems for spacelike hypersurfaces in conformally stationary Lorentz manifolds, one of which not assuming the hypersurface to be of constant mean curvature. Finally, we study the strong r-stability of spacelike hypersurfaces of constant r-th mean curvature in a conformally stationary Lorentz manifold of constant sectional curvature, extending previous results in the current literature.     

Prescribed Mean Curvature Hypersurfaces in Hn+1(-1) with Convex Planar Boundary, I

We study immersed prescribed mean curvature compact hypersurfaces with boundary in Hⁿ⁺¹(-1). When the boundary is a convex planar smooth manifold with all principal curvatures greater than 1, we solve a nonparametric Dirichlet problem and use this, together with a general flux formula, to prove a parametric uniqueness result, in the class of all immersed compact hypersurfaces with the same boundary. We specialize this result to a constant mean curvature, obtaining a characterization of totally umbilic hypersurface caps.     

Enclosure theorems for generalized mean curvature flows

We define a generalized notion of mean curvature for regular hypersurfaces in . This enables us to introduce a new class of geometric curvature flows for which we prove enclosure theorems, using methods of Dierkes [D] and Hildebrandt [H]. In particular, we obtain “neck-pinching” results that generalize previous observations by Ecker [E] concerning the classical mean curvature flow. Received: 8 October 2001 / Accepted: 1 March 2002 / Published online: 23 May 2002     

Harmonic Manifolds and Tubes

Csikós and Horváth (J Lond Math Soc (2) 94(1):141–160, 2016) showed that in a connected locally harmonic manifold, the volume of a tube of small radius about a regularly parameterized simple arc depends only on the length of the arc and the radius. In this paper, we show that this property characterizes harmonic manifolds even if it is assumed only for tubes about geodesic segments. As a consequence, we obtain similar characterizations of harmonic manifolds in terms of the total mean curvature and the total scalar curvature of tubular hypersurfaces about curves. We find simple formulae expressing the volume, total mean curvature, and total scalar curvature of tubular hypersurfaces about a curve in a harmonic manifold as a function of the volume density function.     

Hypersurfaces whose mean curvature function is of finite type

We define finite mean type hypersurfaces to be hypersurfaces with mean curvature function of finite Chen-type. Then, we prove that hyperplanes are the only polynomial translation hypersurfaces of finite mean type in a Euclidean spaceE ⁿ⁺¹. And we show that the only non-conic hyperquadrics of finite mean type in Euclidean spaces are the hyperspheres and the cylinders on spheres. Finally, we state that, among all hypercylinders in a Euclidean spaceE ⁿ⁺¹, the only ones of finite mean type are those on finite mean type planar curves.     

Hypersurfaces with constant anisotropic mean curvature in Riemannian manifolds

We formulate a variational notion of anisotropic mean curvature for immersed hypersurfaces of arbitrary Riemannian manifolds. Hypersurfaces with constant anisotropic mean curvature are characterized as critical points of an elliptic parametric functional subject to a volume constraint. We provide examples of such hypersurfaces in the case of rotationally invariant functionals defined in product spaces. These examples include rotationally invariant hypersurfaces and graphs.     

具有平行平均曲率向量子流形的曲率估计与稳定性

本文估计空间形式中具有平行平均曲率向量子流形上共形度量的数量曲率上界,并利用其研究了具有常平均曲率超曲面的稳定性．     

常平均曲率超曲面的曲率估计与稳定性

本文估计了空间形式Ｎ^ｎ＋１（ｃ）中常平均曲率超曲面上共形度量的曲率上界,并用其研究了Ｎ^ｎ＋１（ｃ）中常平均曲率超曲面的强稳定性．     

Regularity of mean curvature flows with Neumann free boundary conditions

We consider n-dimensional hypersurfaces flowing by the mean curvature flow with Neumann free boundary conditions supported on a smooth support surface. Under assumptions mirroring those for the case of the mean curvature flow without boundary we show that the n-dimensional Hausdorff measure of the singular set is zero.     

Stability of constant mean curvature hypersurfaces of revolution in hyperbolic space

In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stability of these hypersurfaces.