Equivariant min-max hypersurface in $G$-manifolds with positive Ricci curvature

Tongrui Wang

Vol. 331 (2024), No. 1, 149–185

DOI: 10.2140/pjm.2024.331.149

Abstract

We consider a connected orientable closed Riemannian manifold $M^{n+1}$ with positive Ricci curvature. Suppose $G$ is a compact Lie group acting by isometries on $M$ with $3\le \mathrm{codim}<!--FUNCTION\; APPLICATION-->(G\cdot p)\le 7$ for all $p\in M$. Then we show the equivariant min-max $G$-hypersurface $\mathrm{\Sigma }$ corresponding to one-parameter $G$-sweepouts (of boundary-type) is a multiplicity one minimal $G$-hypersurface with a $G$-invariant unit normal and $G$-equivariant index one. As an application, we are able to establish a genus bound for $\mathrm{\Sigma }$, a control on the singular points of $\mathrm{\Sigma }∕G$, and an upper bound for the (first) $G$-width of $M$ provided $n+1=3$ and the actions of $G$ are orientation preserving.

Keywords

min-max theory, equivariant minimal surfaces, positive Ricci curvature, multiplicity one, genus

Mathematical Subject Classification

Primary: 53A10, 53C42

Milestones

Received: 27 April 2023

Revised: 27 June 2024

Accepted: 16 August 2024

Published: 2 October 2024