Equivariant Minimal Surfaces and The Allen-Cahn Equation

For a closed manifold (𝑀,𝑔) with isometry group, 𝐺, we will discuss the construction of 𝐺

-invariant minimal hypersurfaces. Pitts-Rubinstein initiated the program to construct such hypersurfaces via min-max methods. The community has made significant progress when the cohomogeneity of the manifold is at least 3 (i.e. the quotient 𝑀/𝐺

"looks" 3 dimensional). However, serious complications arise when the cohomogeneity is 2, as min-max on closed surfaces produce geodesic networks. Using the Allen-Cahn regularization of the area functional, we provide a general method for constructing minimal hypersurfaces in cohomogeneity 2 when there are no exceptional orbits of the group action, and any cohomogeneity greater than 2 for any action. This is joint work with Pedro Gaspar and Rayssa Caju.