Min-max theory for $G$-invariant minimal hypersurfaces

Abstract

In this paper, we consider a closed Riemannian manifold M n+1 with dimension 3 ≤ n + 1 ≤ 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-invariant version, we show the existence of a nontrivial closed smooth embedded G-invariant minimal hypersurface generated by the min-max procedure. 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[9] and Guth [10]. An application of our results combined with the work of Marques-Neves [22] shows the existence of infinitely many G-invariant minimal hypersurfaces when RicM > 0.

The Existence of G-Invariant constant mean curvature Hypersurfaces

“…We will deal with G-invariant objects in our paper. Now we collect some useful propositions for G-currents and G-varifolds from [30] and [17] .…”

“…Inspired by Ketover's work, a more general version of G-invariant minmax under the smooth sweepouts settings in [6] and [7] was built by Z. Liu in [17] to prove the existence of G-invariant smooth minimal hypersurface on manifolds with dimension 3 ≤ n + 1 ≤ 7. After that, T.R Wang in [30] built the G-equivariant min-max theory under the Almgren-Pitts settings to prove that there are infinitely many minimal hypersurfaces on positive Ricci curvature manifolds with dimension 3 ≤ n + 1 ≤ 7 and dim(M \ M reg ) ≤ n − 2.…”

“…Hence, due to the Ginvariant restrictions, it seems that our variation result is weaker than the classical one which takes all kinds of variations into account. There are already some approaches dealing with this kind of difficulty, see for example [16] [17] [30]. The idea is using an averaging argument to make the usual vector field or current to Ginvariant one.…”

“…To overcome this problem, we show the splitting property of tangent cones as well as various blowups of the c-min-max varifold, which has been used in [17, Appendix 2] and [30,Lemma 6.5]. Due to the splitting property, we only need to show the regularity of tangent cones or various blowups in the normal space N p (G • p).…”

Equivariant index bound for min-max free boundary minimal surfaces

“…Remark 1.3. Here we assume the group G to be finite, but it is due mentioning that the equivariant min-max theory has been developed also in the case of a compact connected Lie group with cohomogeneity not 0 or 2 in [Liu21] and in the Almgren-Pitts setting with a compact Lie group of cohomogeneity greater or equal than 3 in [Wan20].…”