Non-collapsing on Hypersurfaces of Prescribed Curvature

Abstract

The establishment and development of a non-collapsing technique have recently received a great deal of attention in geometric analysis. The area of study conducted in this thesis is motivated by this technique which is based on the application of the maximum principle to a two-point function that is defined on manifolds or relies on some global information of partial differential equations PDEs. This approach is utilized in order to acquire some useful knowledge on the behaviour of geometric structures and properties of embedded hypersurfaces. Despite of that this technique is used in various settings in geometric analysis, it still has a similar machinery in general.

Our first purpose of study concerns the uniqueness of a class of embedded Weingarten hypersurfaces in the higher dimensional sphere $\mathbb{S}^{n+1}$. In particular, we consider a more general class of embedded Weingarten hypersurfaces with two distinct principal curvatures into $\mathbb{S}^{n+1}$. Also, we assume that these hypersurfaces satisfying a PDE of the principal curvatures with assumptions on the number of multiplicities of these curvatures. As a result, we deduce that these principal curvatures are both constant and consequently our hypersurfaces are congruent to a Clifford torus. One of the key ingredients in this work is based on the smart deployment of the maximum principle argument to a function of two variables that is defined on our hypersurfaces.

The second main object of study is the mean convex mean curvature flow in Minkowski space $\mathbb{L}^{n,1}$. Of particular interest is the study of co-compact mean convex embedded spacelike hypersurfaces evolving by the mean curvature flow into $\mathbb{L}^{n,1}$. In particular, we deduce a non-collapsing estimate by employing the maximum principal to a quantity that relies on two points of these spacelike hypersurfaces. More precisely, we compare the radius of the largest hyperbola which touches the spacelike hypersurface at a given point to the curvature at that point. Also, we assume that our spacelike hypersurfaces are asymptotic to the hyperplane in order to to control the behaviour of such hypersurfaces at infinity and ensure that the maximum principle is applicable to our setting. Show more