Chennai Mathematical Institute


Lecture Announcement
Date: Wednesday, August 10, 2022.
Time: 3:30 PM.
Venue: Seminar Hall
Existence of multiple closed CMC hypersurfaces with small mean curvature

Akashdeep Dey
University of Toronto.


The Almgren-Pitts min-max theory deals with the min-max construction of closed minimal hypersurfaces. By the combined works of Almgren, Pitts and Schoen-Simon, every closed Riemannian manifold (M^n , g), n \geq 3, contains at least one closed minimal hypersurface. Recently, the Almgren-Pitts min-max theory has been further developed to show that minimal hypersurfaces exist in abundance. By the works of Marques-Neves and Song, every closed Riemannian manifold (M^n , g), 3 \leq n \leq 7 contains infinitely many closed minimal hypersurfaces. This was conjectured by Yau. Min-max theory for the constant mean curvature (CMC) hypersurfaces has been recently developed by Zhou-Zhu and Zhou. In particular, Zhou and Zhu proved that for any c > 0, every closed Riemannian manifold (M^n , g), n \geq 3, contains a closed c-CMC hypersurface. In the first half of my talk I will briefly discuss the min-max theory for the minimal and CMC hypersurfaces. In the second half, I will talk about the following result. The number of closed c-CMC hypersurfaces in a closed Riemannian manifold (M^n , g), n \geq 3, tends to infinity as c tends to 0^+.