Min-max minimal hypersurfaces with higher multiplicity

TL;DR

This paper constructs examples of non-bumpy metrics on spheres and projective spaces where min-max minimal hypersurfaces have higher multiplicity or specific topological properties, advancing understanding of minimal hypersurface multiplicities.

Contribution

It provides the first examples of non-bumpy metrics with multiplicity-two minimal spheres and explores properties of min-max hypersurfaces in various non-bumpy manifolds.

Findings

Existence of non-bumpy metrics with multiplicity-two minimal spheres on spheres.

Construction of non-bumpy projective spaces with one-sided min-max hypersurfaces.

Examples of non-bumpy balls with improper free boundary min-max hypersurfaces.

Abstract

We exhibit the first set of examples of non-bumpy metrics on the $(n+1)$-sphere ($2\leq n\leq 6$) in which the varifold associated with the two-parameter min-max construction must be a multiplicity-two minimal $n$-sphere. This is proved by a new area-and-separation estimate for certain minimal hypersurfaces with Morse index two inspired by an early work of Colding-Minicozzi. We also construct non-bumpy projective spaces in which the first min-max hypersurfaces are one-sided, and non-bumpy balls in which the free boundary min-max hypersurfaces are improper.