The Weyl problem with nonnegative Gauss curvature in hyperbolic space

TL;DR

This paper addresses the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature, providing bounds and estimates that extend to higher dimensions and include gradient control in two dimensions.

Contribution

The paper introduces new a priori bounds for the second fundamental form and extends embedding results to n-dimensions, also offering gradient estimates in 2D.

Findings

Established bounds for the trace of the second fundamental form H.

Extended isometric embedding results to n-dimensional hyperbolic space.

Derived gradient estimates for the smaller principal curvature in 2D.

Abstract

In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form H and extend the result to n-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions.