Infinite existence of equivariant minimal hypersurfaces
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910130225807360 |
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| author | Li, Xingzhe Wang, Tongrui |
| author_facet | Li, Xingzhe Wang, Tongrui |
| contents | For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show that there are infinitely many $G$-invariant minimal hypersurfaces. Under the assumption that $M$ contains at most a finite number of minimal $G$-hypersurfaces admitting no $G$-invariant unit normal, we further show that each $G$-homology class of $M$ admits infinitely many distinct realizations by embedded minimal $G$-hypersurfaces. The proof relies on a new algorithm that employs multi-stage maximal cuttings. As part of this work, we also established an equivariant min-max theory in manifolds with cylindrical ends. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13422 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Infinite existence of equivariant minimal hypersurfaces Li, Xingzhe Wang, Tongrui Differential Geometry For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show that there are infinitely many $G$-invariant minimal hypersurfaces. Under the assumption that $M$ contains at most a finite number of minimal $G$-hypersurfaces admitting no $G$-invariant unit normal, we further show that each $G$-homology class of $M$ admits infinitely many distinct realizations by embedded minimal $G$-hypersurfaces. The proof relies on a new algorithm that employs multi-stage maximal cuttings. As part of this work, we also established an equivariant min-max theory in manifolds with cylindrical ends. |
| title | Infinite existence of equivariant minimal hypersurfaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2604.13422 |