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- Title
Equivariant Morse index of min–max G-invariant minimal hypersurfaces.
- Authors
Wang, Tongrui
- Abstract
For a closed Riemannian manifold M n + 1 with a compact Lie group G acting as isometries, the equivariant min–max theory gives the existence and the potential abundance of minimal G-invariant hypersurfaces provided 3 ≤ codim (G · p) ≤ 7 for all p ∈ M . In this paper, we show a compactness theorem for these min–max minimal G-hypersurfaces and construct a G-invariant Jacobi field on the limit. Combining with an equivariant bumpy metrics theorem, we obtain a C G ∞ -generic finiteness result for min–max G-hypersurfaces with area uniformly bounded. As a main application, we further generalize the Morse index estimates for min–max minimal hypersurfaces to the equivariant setting. Namely, the closed G-invariant minimal hypersurface Σ ⊂ M constructed by the equivariant min–max on a k-dimensional homotopy class can be chosen to satisfy Index G (Σ) ≤ k .
- Subjects
HYPERSURFACES; COMPACT groups; RIEMANNIAN manifolds; LIE groups; INVARIANT sets
- Publication
Mathematische Annalen, 2024, Vol 389, Issue 2, p1599
- ISSN
0025-5831
- Publication type
Article
- DOI
10.1007/s00208-023-02681-z