Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: Morse theory for the area functional
Speaker: Akashdeep Dey, Princeton University and University of Toronto
Date: October 10, 2022
Time: 16:30:00 Hours
Venue: AG-77
Abstract: This is the second talk in a series of three talks on the variational theory of minimal hypersurfaces. In this talk, I will discuss the following theorems. When the ambient dimension $3 \leq n \leq 7$, Marques and Neves showed that the index of the min-max minimal hypersurface is bounded from above by the dimension of the parameter space. Zhou proved the multiplicity one property for the min-max minimal hypersurfaces, which was conjectured by Marques and Neves. In the Almgren-Pitts min-max theory, the min-max width is realized by the area of a closed minimal hypersurface, with the possibility that the connected components of the minimal hypersurface can have different multiplicities. The multiplicity one theorem says that for a generic metric, all the min-max minimal hypersurfaces have multiplicity one. Using the Morse index upper bound and multiplicity one theorem, Marques and Neves have proved the following theorem. For a generic metric $g$, there exists a sequence of closed, embedded, two-sided minimal hypersurfaces ${S_p}$ in $(M^n, g)$ such that the Morse index Ind$(S_p) = p$ and area$(S_p) \sim p^{1/n}$. In higher dimensions (i.e. when $n \geq 8)$, the Morse index upper bound has been proved by Li.
Title of Talk: Morse theory for the area functional
Speaker: Akashdeep Dey, Princeton University and University of Toronto
Date: October 10, 2022
Time: 16:30:00 Hours
Venue: AG-77
Abstract: This is the second talk in a series of three talks on the variational theory of minimal hypersurfaces. In this talk, I will discuss the following theorems. When the ambient dimension $3 \leq n \leq 7$, Marques and Neves showed that the index of the min-max minimal hypersurface is bounded from above by the dimension of the parameter space. Zhou proved the multiplicity one property for the min-max minimal hypersurfaces, which was conjectured by Marques and Neves. In the Almgren-Pitts min-max theory, the min-max width is realized by the area of a closed minimal hypersurface, with the possibility that the connected components of the minimal hypersurface can have different multiplicities. The multiplicity one theorem says that for a generic metric, all the min-max minimal hypersurfaces have multiplicity one. Using the Morse index upper bound and multiplicity one theorem, Marques and Neves have proved the following theorem. For a generic metric $g$, there exists a sequence of closed, embedded, two-sided minimal hypersurfaces ${S_p}$ in $(M^n, g)$ such that the Morse index Ind$(S_p) = p$ and area$(S_p) \sim p^{1/n}$. In higher dimensions (i.e. when $n \geq 8)$, the Morse index upper bound has been proved by Li.