Frankel property and Maximum Principle at Infinity for complete minimal hypersurfaces

TL;DR

This paper extends the maximum principle at infinity and the Frankel property to higher-dimensional complete minimal hypersurfaces in Riemannian manifolds, revealing new geometric bounds and classifications.

Contribution

It generalizes known results from three dimensions to dimensions 4 through 7, including a maximum principle at infinity and a classification of stable minimal hypersurfaces.

Findings

Disjoint minimal hypersurfaces bound a product slab in certain manifolds.

Generalization of Mazet's maximum principle to higher dimensions.

Classification of stable minimal hypersurfaces in higher dimensions.

Abstract

In this paper, we study complete minimal hypersurfaces in Riemannian $n-$manifolds $\mathcal{M}^n$ for dimensions $4 \leq n \leq 7$, and we obtain some results in the spirit of known work for $n=3$. Key contributions include extending the work of Anderson and Rodr\'{i}guez to higher dimensions. Specifically, we show that in four-dimensional manifolds with nonnegative sectional curvature and positive scalar curvature, two disjoint properly embedded minimal hypersurfaces bound a slab isometric to the product of one hypersurface with an interval. Our results are grounded in a maximum principle at infinity for two-sided, parabolic, properly embedded minimal hypersurfaces in complete Riemannian manifolds of bounded geometry, generalizing the work of Mazet in dimension three to higher dimensions. We also leverage the recent classification of complete two-sided stable minimal hypersurfaces by Chodosh, Li, and Stryker.