论文标题
$ g $ invariant的最小超曲面的最小最大理论
Min-max theory for $G$-invariant minimal hypersurfaces
论文作者
论文摘要
在本文中，我们考虑了一个封闭的Riemannian歧管$ M^{n+1} $，dimension $ 3 \ leq n+1 \ leq 7 $，以及一个紧凑型谎言组$ g $，在$ m $的同量值上，带有cohomogeneity，至少$ 3 $。在将Almgren-Pitts Min-Max理论调整为$ g $ equivariant版本之后，我们表明存在非平凡的平滑光滑嵌入式$ g $ g $ g $ invariant的最小超出表面$σ\ subset m $，规定非原始轨道的结合了非原始轨道的结合，是$ n $ n $ n $ n dim $ n dim $ n-dimbess $ n dim $ n dim $ n dim $ n dim $ n dim $ n dim dim dim dim的，此外，我们还建立了上限以及$（g，p）$宽度的下限，这是Gromov和Guth得出的经典结论的类似物。我们的结果的应用与Marques-neves的工作相结合，表明存在无限的$ G $ - invariant最小超曲面时，当$ {\ rm rm ric} _m> 0 $时，Orbits满足了上面的同一假设。
In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries on $M$ with cohomogeneity at least $3$. After adapting the Almgren-Pitts min-max theory to a $G$-equivariant version, we show the existence of a nontrivial closed smooth embedded $G$-invariant minimal hypersurface $Σ\subset M$ provided that the union of non-principal orbits forms a smooth embedded submanifold of $M$ with dimension at most $n-2$. Moreover, we also build upper bounds as well as lower bounds of $(G,p)$-width which are analogs of the classical conclusions derived by Gromov and Guth. An application of our results combined with the work of Marques-Neves shows the existence of infinitely many $G$-invariant minimal hypersurfaces when ${\rm Ric}_M>0$ and orbits satisfy the same assumption above.
