Min–Max Theory for G-Invariant Minimal Hypersurfaces

Min-max theory for free boundary G-invariant minimal hypersurfaces

“…Moreover, if ∂M = ∅, Theorem 1.2 provides a closed embedded minimal G-invariant hypersurface. Therefore, our theorem not only extends the results in [16] [20] [32] into the free boundary case, but also weakens the requirements for Lie groups.…”

“…The following theorem shows we can generate an M-continuous map from a discrete map with small M-fineness, which is parallel to [19,Theorem 4.14] and [32,Theorem 3]. We call the map Φ in Theorem 4.13 the Almgren G-extension of φ.…”

“…where T i ∈ τ i , i ∈ {1, 2} are the canonical representatives. By Illman's work [13][14], we can get a triangulation △ of the orbit space M/G, which generates an equivariant triangulation △ := {π −1 (s) : s ∈ △} of M with some nice properties (see [32,Appendix A]). Therefore, we can construct the cutting maps and the deforming maps for integral G-currents same as in [32,Theorem 3].…”

“…[6]) using generalized hypersurfaces instead of currents. Therefore, the author proposed in [32] an equivariant min-max construction under the setting of Almgren-Pitts, where M n+1 is assumed to be closed with dimension 3 ≤ n + 1 ≤ 7, G is assumed to be a compact Lie group with Cohom(G) ≥ 3, and M \ M reg is assumed to be a submanifold of M with dimension no more than n−2. Here M reg is the union of all principal orbits, and Cohom(G) is the cohomogeneity of the action of G.…”

“…Nevertheless, if every orbit in the compact G-set Clos(U ) has the same orbit type, then there exists a uniform injectivity radius of orbits in Clos(U ). Therefore, the author add a technical assumption in [32] that M \ M reg is a submanifold with dimension no more than n − 2, which guarantees the closure of the small G-annulus around M \ M reg is contained in M reg , and thus the interpolation arguments in [26, ] would carry over with exp ⊥ G•p in place of exp p (see [32, Theorem 5.5, Appendix B]). Inspired by the work of Li-Zhou [19, Appendix B], we can now overcome this technical problem without any further assumptions on M \ M reg .…”