论文标题
Min-Max $ g $ g $ invariant minimar timimal hypersurfaces的模棱两可的摩尔斯索引指数
Equivariant Morse index of min-max $G$-invariant minimal hypersurfaces
论文作者
论文摘要
对于带有紧凑型谎言组$ g $的封闭式riemannian歧管$ m^{n+1} $起作用的lie copt $ g $，等效的min-max理论提供了最小$ g $ g $ g $ g $ g $ invariant hypersurfaces的潜在丰富性，提供了$ 3 \ leq {\ rm codim} $ codim}（g codim}（for $ cdot p p p p p p）在本文中，我们显示了这些最小的$ g $ hypersurfaces的紧凑定理，并在极限上构建了$ g $ invariant jacobi领域。结合一个颠簸的度量定理，我们获得了$ c^\ infty_g $ - 基因的有限性结果，用于Min-Max $ g $ -Hypersurfaces，区域均匀界限。作为主要应用程序，我们进一步概括了MIN-MAX最小超曲面的Morse指数估计值。也就是说，可以选择封闭的$ g $ - invariant最小的超曲面$σ\子集M $由Equivariant Min-Max在$ k $ - 二维同质类上构建的，可以选择满足$ {\ rm index} _g（σ）\ leq k $。
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 $3\leq {\rm codim}(G\cdot p) \leq 7$ for all $p\in M$. 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 $C^\infty_G$-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 $Σ\subset M$ constructed by the equivariant min-max on a $k$-dimensional homotopy class can be chosen to satisfy ${\rm Index}_G(Σ)\leq k$.
