Analysis and Mathematical Physics

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.