Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces

TL;DR

This paper establishes new criteria for criticality and spectral splitting theorems for operators on manifolds with multiple ends, and applies these to classify stable minimal hypersurfaces in low-dimensional ambient spaces.

Contribution

It introduces general criticality criteria and spectral splitting theorems under spectral Ricci bounds, extending classical results and providing new classifications of minimal hypersurfaces.

Findings

A 1/3-stable minimal hypersurface in R^4 is either one-ended or a catenoid.

Proper δ-stable minimal hypersurfaces with δ > 1/3 in R^4 are hyperplanes.

New insights into manifolds with weighted Poincaré inequalities and their spectral properties.

Abstract

In this paper we prove general criticality criteria for operators $\Delta + V$ on manifolds with more than one end, where $V$ bounds the Ricci curvature, and a related spectral splitting theorem extending Cheeger-Gromoll's one. Our results give new insight on Li-Wang's theory of manifolds with a weighted Poincar\'e inequality. We apply them to study stable and $\delta$-stable minimal hypersurfaces in manifolds with non-negative bi-Ricci or sectional curvature, in ambient dimension up to $5$ and $6$, respectively. In the special case where the ambient space is $\mathbb{R}^4$, we prove that a $1/3$-stable minimal hypersurface must either have one end or be a catenoid, and that proper, $\delta$-stable minimal hypersurfaces with $\delta > 1/3$ must be hyperplanes.