Spectral splitting theorem and ends of minimal hypersurfaces

TL;DR

This paper provides a new proof of the spectral splitting theorem and shows that minimal hypersurfaces with finite index in certain manifolds have finitely many ends, extending previous results.

Contribution

It introduces a novel proof of the spectral splitting theorem and generalizes the finite ends property for minimal hypersurfaces in manifolds with nonnegative biRic curvature.

Findings

New proof of the spectral splitting theorem for manifolds with nonnegative spectral Ricci curvature.

Minimal hypersurfaces with finite index in manifolds with nonnegative biRic curvature have finitely many ends.

Extends Li-Wang's result to broader curvature conditions.

Abstract

In this paper, we give a new proof of the splitting theorem on manifolds with nonnegative spectral Ricci curvature proved in [APX24, CMMR24, HW26]. Furthermore, by constructing weighted minimizing geodesics at infinity, we show that minimal hypersurfaces with finite index in manifolds with nonnegative biRic curvature must have finite ends, generalizing the result of Li-Wang [LW04] on manifolds with nonnegative sectional curvature.