Mean Curvature Function Research Articles

A characterization of compact surfaces with constant mean curvature

Surfaces in a 3 3 -space of constant curvature, whose arbitrary sufficiently small open subsets admit a non-trivial isometric deformation preserving the mean curvature function, are called locally H H -deformable. It is well known that surfaces with constant mean curvature which are not totally umbilical are all locally H H -deformable. Conversely, we shall show in this paper that any compact locally H H -deformable surface has constant mean curvature.

Immersed hypersurfaces with constant Weingarten curvature

In a recent paper Korevaar 1-5] used the Alexandrov reflection principle to show that closed embedded hypersurfaces in N 1, lHn+ 1 or the upper hemisphere of S~ + 1 are umbilic spheres provided a certain function f of the principal curvatures ,t=().t,22 .. . . ,)~,) is constant. He only had to assume that f is positive on the positive cone ~={212i>0 Vi} and that f is elliptic on the component F of {21 f(,~)>0} which contains ~. Here f is said to be elliptic if 0f/d21>0 for all i, 1 <iNn . This generalization of earlier sphere theorems (see [9] and [10] for references) cannot be extended to hypersurface immersions in view of recent counterexamples, 1-11]. However, assuming additional curvature conditions Walter derived in [10] global results for hypersurface immersions in a space N~+l(c) of constant curvature c, which have a constant higher mean curvature function H,. Here H, is the r-th symmetric function of the principal curvatures. It was shown that such hypersurfaces are of constant mean curvature H1, provided they have nonnegative sectional curvature and non-negative principal curvatures. As a consequence they have to be isoparametric with at most two distinct principal curvatures and can therefore be completely classified. Here we show that it is not necessary to assume all principal curvatures to be non-negative. Moreover we extend Waiter's result to general symmetric functions f = f(2). Let M ~ be a smooth, connected and compact manifold without boundary. Then we have the following result.

Isometric deformation of surfaces inR3preserving the mean curvature function

The purpose of this paper is to classify surfaces in Euclidean 3space with constant Gaussian curvature which admit non-trivial oneparameter families of isometric immersions preserving the mean curvature function. It is shown that the Gaussian curvature must be zero and, if the mean curvature is not constant, then such isometric immersions are some deformations of the cylinder over a logarithmic spiral.

On the Gaussian and mean curvatures of parallel hypersurfaces

In this study, we give a generaiizatîon of the higher order Gaussian curvature function and the mean curvature function ahout the parallel surfaces in E^, for E^’^*.

Discrimination of contour curvature: data and theory.

Visual discrimination of contour curvature was investigated using stimuli having a single point of maximum curvature and a continuous derivative. Curvature discrimination as a function of mean curvature could be described by a power law with an exponent averaging 1.57. Data were also gathered as a function of line width, stimulus orientation, and retinal eccentricity. Finally, masking experiments provided evidence that the mechanisms responsible for curvature discrimination were both orientation and spatial-frequency selective. The data are well fitted by theoretical results derived from a line-element model that has recently been applied to spatial-frequency discrimination.

On isometric immersions of a torus into a space form with the same mean curvature function

On isometric immersions of a torus into a space form with the same mean curvature function

Compact hypersurfaces with a constant higher mean curvature function

Compact hypersurfaces with a constant higher mean curvature function

Mean curvature functions for codimension-one foliations with all leaves compact

Mean curvature functions for codimension-one foliations with all leaves compact

On the higher mean curvature functions of Kaehler hypersurfaces in complex space forms

On the higher mean curvature functions of Kaehler hypersurfaces in complex space forms