Index and topology of minimal hypersurfaces in R^n

TL;DR

This paper establishes lower bounds on the Morse index and nullity of minimal hypersurfaces in Euclidean space with finite total curvature, leading to new compactness and finiteness results in four dimensions.

Contribution

It provides the first effective bounds relating index, nullity, ends, and topology of minimal hypersurfaces, extending Li-Wang's work and applying to compactness in $R^4$.

Findings

Lower bounds on Morse index and nullity in terms of ends and Betti number.

Finiteness and compactness results for minimal hypersurfaces in $R^4$ with finite index.

Abstract

In this paper, we consider immersed two-sided minimal hypersurfaces in $\mathbb{R}^n$ with finite total curvature. We prove that the sum of the Morse index and the nullity of the Jacobi operator is bounded from below by a linear function of the number of ends and the first Betti number of the hypersurface. When $n=4$, we are able to drop the nullity term by a careful study for the rigidity case. Our result is the first effective generalization of Li-Wang. Using our index estimates and ideas from the recent work of Chodosh-Ketover-Maximo, we prove compactness and finiteness results of minimal hypersurfaces in $\mathbb{R}^4$ with finite index.