Equivariant min-max hypersurface in $G$-manifolds with positive Ricci curvature
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910637967278080 |
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| author | Wang, Tongrui |
| author_facet | Wang, Tongrui |
| contents | In this paper, 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 $3\leq {\rm codim}(G\cdot p)\leq 7$ for all $p\in M$. Then we show the equivariant min-max $G$-hypersurface $Σ$ corresponding to the fundamental class $[M]$ 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 $Σ$, a control on the singular points of $Σ/G$, and an upper bound for the (first) $G$-width of $M$ provided $n+1=3$ and the actions of $G$ are orientation preserving. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_03656 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Equivariant min-max hypersurface in $G$-manifolds with positive Ricci curvature Wang, Tongrui Differential Geometry In this paper, 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 $3\leq {\rm codim}(G\cdot p)\leq 7$ for all $p\in M$. Then we show the equivariant min-max $G$-hypersurface $Σ$ corresponding to the fundamental class $[M]$ 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 $Σ$, a control on the singular points of $Σ/G$, and an upper bound for the (first) $G$-width of $M$ provided $n+1=3$ and the actions of $G$ are orientation preserving. |
| title | Equivariant min-max hypersurface in $G$-manifolds with positive Ricci curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2304.03656 |