Existence and non-existence of area-minimizing hypersurfaces in manifolds of non-negative Ricci curvature

TL;DR

This paper investigates the conditions under which complete area-minimizing hypersurfaces exist in manifolds with non-negative Ricci curvature, identifying a sharp decay threshold that determines existence or non-existence.

Contribution

It establishes a precise decay rate of Ricci curvature that acts as a borderline for the existence of area-minimizing hypersurfaces in such manifolds.

Findings

No complete area-minimizing hypersurfaces exist above the decay threshold.

Examples of hypersurfaces exist below the decay threshold.

The decay rate is characterized by comparison with capped spherical cones.

Abstract

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples.