# Geometry Seminar: Existence of minimal hypersurfaces with arbitrarily large area and Morse index

## Location

The recent development of Almgren-Pitts min-max theory has presented the abundance of minimal hypersurfaces. In particular, in a bumpy closed Riemannian manifold $(M^{n+1}, g)$ $(3\leq n+1\leq 7)$, X. Zhou’s multiplicity one theorem and Marques-Neves Morse index theorem lead to the fact that for each positive integer $p$, there exists a minimal hypersurface with Morse index $p$ and area propotional to $p^{1/(n+1)}$. However, the minimal hypersurface here might have multiple disjoint connected components, so the geometric complexity of it could merely be the accumulation of its components with relatively small area or Morse index, and it is not clear whether at least one component would have such geometric complexity. In this talk, by adapting the theorems mentioned above into a confined min-max setting, we will show that such a manifold always admits a sequence of connected closed embedded two-sided minimal hypersurfaces whose areas and Morse indices both tend to infinity. This improves a previous result by O. Chodosh and C. Mantoulidis on connected minimal hypersurfaces with arbitrarily large area.