# Curvature estimates for Weingarten hypersurfaces in Riemannian manifolds

## TL;DR

This paper establishes curvature estimates for general curvature functions and demonstrates the existence of convex hypersurfaces with prescribed curvature in Riemannian manifolds, under broad conditions.

## Contribution

It provides new curvature estimates for a wide class of curvature functions and proves the existence of convex hypersurfaces with prescribed curvature in Riemannian manifolds.

## Key findings

- Curvature estimates for general curvature functions are proven.
- Existence of closed, strictly convex hypersurfaces with prescribed curvature is shown.
- Applicable to curvature functions that are monotone, symmetric, homogeneous, concave, and smooth.

## Abstract

We prove curvature estimates for general curvature functions. As an application we show the existence of closed, strictly convex hypersurfaces with prescribed curvature $F$, where the defining cone of $F$ is $\C_+$. $F$ is only assumed to be monotone, symmetric, homogeneous of degree 1, concave and of class $C^{m,\al}$, $m\ge4$.