Hessian curvature hypersurfaces with prescribed Gauss image

TL;DR

This paper establishes the existence of convex hypersurfaces with prescribed k-Hessian curvatures and Gauss images, using a novel boundary estimate based on orthogonal invariance and infinitesimal rotations.

Contribution

Introduces a new $C^2$ boundary estimate technique leveraging hypersurface invariance and rotation vector fields for prescribed curvature problems.

Findings

Existence of convex hypersurfaces with prescribed Gauss images and k-Hessian curvatures.

Development of a novel boundary $C^2$ estimate method.

Handling of negative boundary terms via orthogonal invariance.

Abstract

In this paper, we investigate Hessian curvature hypersurfaces with prescribed Gauss images. Given geodesically strictly convex bounded domains $\Omega$ in $\mathbb{R}^n$ and $\tilde{\Omega}$ in the unit hemisphere, we prove that there is a strictly convex graphic hypersurface defined in $\Omega$ with prescribed $k$-Hessian curvatures such that its Gauss image is $\tilde{\Omega}$. Our proof relies on a novel $C^2$ boundary estimate which utilizes the orthogonal invariance of hypersurfaces. Indeed, we employ some special vector fields generated by the infinitesimal rotations in $\mathbb{R}^{n+1}$ to establish the boundary $C^2$ estimates. This new approach enables us to handle the additional negative terms that arise when taking second order derivatives near the boundary.