Starshaped compact hypersurfaces in warped product manifolds I: prescribed curvature equations

TL;DR

This paper develops global curvature estimates for star-shaped hypersurfaces in warped product manifolds satisfying prescribed curvature equations, extending to semi-convex and $k$-convex solutions for broader classes of curvature problems.

Contribution

It introduces a unified approach to derive curvature estimates for various convexity conditions and curvature equations in warped product manifolds, broadening the scope of geometric analysis techniques.

Findings

Established curvature estimates for $(n-2)$-convex hypersurfaces.

Extended estimates to semi-convex solutions of $k$-curvature equations.

Applied results to prescribed curvature measure type equations.

Abstract

We derive global curvature estimates for closed, strictly star-shaped $(n-2)$-convex hypersurfaces in warped product manifolds, which satisfy the prescribed $(n-2)$-curvature equation with a general right-hand side. The proof can be readily adapted to establish curvature estimates for semi-convex solutions to the general $k$-curvature equation. Furthermore, it can also be used to prove the same estimates for $k$-convex solutions to the prescribed curvature measure type equations.