A locally constrained inverse Hessian quotient flow in de Sitter space

TL;DR

This paper introduces a new locally constrained inverse curvature flow in de Sitter space, establishing an Alexandrov-Fenchel inequality for 2-convex spacelike hypersurfaces, partially addressing a conjecture by Hu and Li.

Contribution

It develops a novel curvature flow approach to prove an Alexandrov-Fenchel inequality in de Sitter space, advancing geometric analysis in Lorentzian manifolds.

Findings

Established an Alexandrov-Fenchel inequality for 2-convex hypersurfaces in de Sitter space.

Demonstrated the convergence behavior of the locally constrained inverse curvature flow.

Provided partial confirmation of Hu and Li's conjecture.

Abstract

In this paper, an Alexandrov-Fenchel inequality is established for closed $2$-convex spacelike hypersurface in de Sitter space by investigating the behavior of the locally constrained inverse curvature flow \begin{align} \frac{\partial }{\partial t}x=\bigg(u-\frac{\lambda'E_1}{E_2}\bigg)\nu,\nonumber \end{align} which provides a partial answer to the conjecture raised by Hu and Li in \cite{HL}.