Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: Abundant existence of minimal hypersurfaces
Speaker: Akashdeep Dey, Princeton University and University of Toronto
Date: October 17, 2022
Time: 16:30:00 Hours
Venue: AG-77
Abstract: This is the third talk in a series of three talks on the variational theory of minimal hypersurfaces. In this talk, I will discuss the following theorems. By the works of Marques-Neves and Song, every closed Riemannian manifold $M^n, 3 \leq n \leq 7$, contains infinitely many closed, minimal hypersurfaces. This was conjectured by Yau. For generic metrics, stronger results have been obtained. Irie, Marques and Neves proved that for a generic metric $g$ on $M$, the union of all closed, minimal hypersurfaces is dense in $(M, g)$. This theorem was later quantified by Marques, Neves and Song; they proved that for a generic metric there exists an equidistributed sequence of closed, minimal hypersurfaces in $(M, g)$. In higher dimensions, Li proved that every closed Riemannian manifold, equipped with a generic metric, contains infinitely many closed minimal hypersurfaces. The Weyl law for the volume spectrum, proved by Liokumovich, Marques and Neves, played a major role in the proofs of these theorems. Inspired by the abundant existence of closed minimal hypersurfaces, we showed that the number of closed $c$-CMC hypersurfaces in a closed Riemannian manifold $M^n, n \geq 3$, tends to infinity as $c \rightarrow 0^+$.
Title of Talk: Abundant existence of minimal hypersurfaces
Speaker: Akashdeep Dey, Princeton University and University of Toronto
Date: October 17, 2022
Time: 16:30:00 Hours
Venue: AG-77
Abstract: This is the third talk in a series of three talks on the variational theory of minimal hypersurfaces. In this talk, I will discuss the following theorems. By the works of Marques-Neves and Song, every closed Riemannian manifold $M^n, 3 \leq n \leq 7$, contains infinitely many closed, minimal hypersurfaces. This was conjectured by Yau. For generic metrics, stronger results have been obtained. Irie, Marques and Neves proved that for a generic metric $g$ on $M$, the union of all closed, minimal hypersurfaces is dense in $(M, g)$. This theorem was later quantified by Marques, Neves and Song; they proved that for a generic metric there exists an equidistributed sequence of closed, minimal hypersurfaces in $(M, g)$. In higher dimensions, Li proved that every closed Riemannian manifold, equipped with a generic metric, contains infinitely many closed minimal hypersurfaces. The Weyl law for the volume spectrum, proved by Liokumovich, Marques and Neves, played a major role in the proofs of these theorems. Inspired by the abundant existence of closed minimal hypersurfaces, we showed that the number of closed $c$-CMC hypersurfaces in a closed Riemannian manifold $M^n, n \geq 3$, tends to infinity as $c \rightarrow 0^+$.