Curvature estimates for minimal hypersurfaces via generalized longitude function

TL;DR

This paper introduces a generalized longitude function on convex supporting sets of spheres, leading to new curvature estimates and Bernstein theorems for minimal hypersurfaces by analyzing harmonic maps and their regularity.

Contribution

It develops a novel generalized longitude function with totally geodesic level sets and applies it to derive curvature bounds and regularity results for minimal hypersurfaces.

Findings

Established image shrinking property for harmonic maps

Derived curvature estimates for minimal hypersurfaces

Proved Bernstein-type theorems using the new approach

Abstract

On some specified convex supporting sets of spheres, we find a generalized longitude function whose level sets are totally geodesic. Given an arbitrary (weakly) harmonic map into spheres, the composition of the generalized longitude function and harmonic map satisfies an elliptic equation of divergence type. With the aid of corresponding Harnack inequality, we establish image shrinking property and then the regularity results are followed. Applying such results to study the Gauss image of minimal hypersurfaces in Euclidean spaces, we obtain curvature estimates and corresponding Bernstein theorems.