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arXiv · 2208.07187

Min-max theory for free boundary G-invariant minimal hypersurfaces

Tongrui Wang
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Abstract

Given a compact Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$ and $\partial M\neq\emptyset$, the free boundary min-max theory built by Martin Man-Chun Li and Xin Zhou shows the existence of a smooth almost properly embedded minimal hypersurface with free boundary in $\partial M$. In this paper, we generalize their constructions into equivariant settings. Specifically, let $G$ be a compact Lie group acting as isometries on $M$ with cohomogeneity at least $3$. Then we show that there exists a nontrivial smooth almost properly embedded $G$-invariant minimal hypersurface with free boundary. Moreover, if the Ricci curvature of $M$ is non-negative and $\partial M$ is strictly convex, then there exist infinitely many properly embedded $G$-invariant minimal hypersurfaces with free boundary.

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Tongrui Wang. 2022-08-15. Min-max theory for free boundary G-invariant minimal hypersurfaces. https://arxiv.org/abs/2208.07187

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