On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli, Setti (2005) [17].     

Surfaces with harmonic inverse mean curvature in space forms

We define surfaces with harmonic inverse mean curvature in space forms and generalize a theorem due to Lawson by which surfaces of constant mean curvature in one space form isometrically correspond to those in another. We also obtain an immersion formula, which gives a deformation family for these surfaces.

     

Parallel mean curvature biconservative surfaces in complex space forms

In this paper, we consider PMC surfaces in complex space forms, and study the interaction between the notions of PMC, totally real and biconservative. We first consider PMC surfaces in a non-flat complex space form and prove that they are biconservative if and only if totally real. Then, we find a Simons-type formula for a well-chosen vector field constructed from the mean curvature vector field and use it to prove a rigidity result for CMC biconservative surfaces in two-dimensional complex space forms. We prove then a reduction codimension result for PMC biconservative surfaces in non-flat complex space forms. We conclude by constructing examples of CMC non-PMC biconservative submanifolds from the Segre embedding and discuss when they are proper-biharmonic.     

An estimate for the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we derive a sharp estimate for the supremum of the scalar curvature (or, equivalently, the infimum of the squared norm of the second fundamental form) of a constant mean curvature hypersurface with two principal curvatures immersed into a Riemannian space form of constant curvature. Our results will be an application of the generalized Omori-Yau maximum principle, following the approach by Pigola et al. (Memoirs Am Math Soc 822, 2005).     

Topological change in mean convex mean curvature flow

Consider the mean curvature flow of an (n+1)-dimensional compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the mth homotopy group of the complementary region can die only if there is a shrinking S ^k ×R ^n?k singularity for some k≤m. We also prove that for each m with 1≤m≤n, there is a nonempty open set of compact, mean convex regions K in R ⁿ⁺¹ with smooth boundary ?K for which the resulting mean curvature flow has a shrinking S ^m ×R ^n?m singularity.     

Compact Hopf hypersurfaces of constant mean curvature in complex space forms

We prove that every connected compact Hopf hypersurface of a complex space form, contained in a geodesic ball of radius strictly smaller than the injectivity radius of, having constant mean curvature and with if if < 0 is a geodesic sphere of.Work partially supported by DGICYT Grant No. PB91-0324.     

Biminimal Lagrangian surfaces of constant mean curvature in complex space forms

We determine all biminimal Lagrangian surfaces of non-zero constant mean curvature in 2-dimensional complex space forms.     

The mean integral curvature and infinitesimal deformations of a surface in a three-dimensional Riemannian space

We find a formula for the derivative of the mean integral curvature of a surface in a three-dimensional Riemannian space with respect to infinitesimal deformations.     

Strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms

Abstract

This article is devoted to C² a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature equation under the assumption of a strictly locally convex subsolution.     

Stability of mean convex cones under mean curvature flow

We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature. If another solution is initially close to the cone at infinity, then the difference to the homothetically expanding solution becomes small for large times. The proof involves the construction of appropriate barriers.     

On the existence of integral currents with prescribed mean curvature vector

Given an integralm-currentT ₀ in ℝ ^m+k and a tensorH of typ (m, 1) on ℝ ^m+k with values orthogonal to each of its arguments we prove the existence of an integralm-currentT with boundary ∂T=∂T ₀ having prescribed mean curvature vectorH, i. e. is a solution of      

Forced convex mean curvature flow in Euclidean spaces

We show the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term may shrink to a point in finite time if the forcing term is small, or exist for all times and expand to infinity if the forcing term is large enough. The flow can converge to a round sphere in special cases. Long time existence and convergence of the normalization of the flow are studied.