Curvature estimates for submanifolds with prescribed Gauss image and mean curvature

TL;DR

This paper establishes interior curvature estimates for n-graphs with prescribed mean curvature and Gauss image constraints, extending classical results and providing new bounds under various geometric conditions.

Contribution

It introduces novel interior curvature estimates for submanifolds with prescribed mean curvature and Gauss image restrictions, including cases without dimension limitations.

Findings

Derived curvature bounds under dimension restrictions.

Extended estimates without dimension limitations involving the Gauss map.

Provided curvature estimates when the Gauss image lies in a geodesic ball.

Abstract

We study that the $n-$graphs defining by smooth map $f:\Om\subset \ir{n}\to \ir{m}, m\ge 2,$ in $\ir{m+n}$ of the prescribed mean curvature and the Gauss image. We derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le\f{C}{R^2}$$ under the dimension limitations and the Gauss image restrictions. If there is no dimension limitation we obtain $$\sup_{D_R(x)}|B|^2\le C R^{-a}\sup_{D_{2R}(x)}(2-\De_f)^{-(\f{3}{2}+\f{1}{s})}, \qquad s=\min(m, n)$$ with $a<1$ under the condition $$\De_f=[\text{det}(\de_{ij}+\sum_\a\pd{f^\a}{x^i}\pd{f^\a}{x^j})]^{\f{1}{2}}<2.$$ If the image under the Gauss map is contained in a geodesic ball of the radius $\f{\sqrt{2}}{4}\pi$ in $\grs{n}{m}$ we also derive corresponding estimates.