Variational Methods in Geometry

Multiplicity One Conjecture in Min-max theory

Xin Zhou
University of California, Santa Barbara; Member, School of Mathematics
March 19, 2019

I will present a proof with some substantial details of the Multiplicity One Conjecture in Min-max theory, raised by Marques and Neves. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are all two-sided and have multiplicity one. 

Macroscopically minimal hypersurfaces

Hannah Alpert
Ohio State University
March 12, 2019

A decades-old application of the second variation formula
proves that if the scalar curvature of a closed 3--manifold is bounded
below by that of the product of the hyperbolic plane with the line,
then every 2--sided stable minimal surface has area at least that of
the hyperbolic surface of the same genus. We can prove a coarser
analogue of this statement, taking the appropriate notions of
macroscopic scalar curvature and macroscopic minimizing hypersurface
from Guth's 2010 proof of the systolic inequality for the

Geodesic nets: examples and open problems.

Alexander Nabutovsky
University of Toronto; Member, School of Mathematics
February 26, 2019

Geodesic nets on Riemannian manifolds is a natural generalization of geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round 2-sphere. 

In this talk I am going to survey some results and open questions (old
and new) about geodesic nets on Riemannian manifolds. Many of these

On minimizers and critical points for anisotropic isoperimetric problems

Robin Neumayer
Member, School of Mathematics
February 19, 2019

Anisotropic surface energies are a natural generalization of the perimeter functional that arise in models in crystallography and in scaling limits for certain probabilistic models on lattices. This talk focuses on two results concerning isoperimetric problems with anisotropic surface energies. In the first part of the talk, we will discuss a weak characterization of critical points in the anisotropic isoperimetric problem (joint work with Delgadino, Maggi, and Mihaila). 

Isoperimetry and boundaries with almost constant mean curvature

Francesco Maggi
The University of Texas at Austin; Member, School of Mathematics
February 12, 2019
We review various recent results aimed at understanding bubbling into spheres for boundaries with almost constant mean curvature. These are based on joint works with Giulio Ciraolo (U Palermo), Matias Delgadino (Imperial College London), Brian Krummel (Purdue), Cornelia Mihaila (U Chicago), and Robin Neumayer (Nothwestern and IAS).

Min-max solutions of the Ginzburg-Landau equations on closed manifolds

Daniel Stern
Princeton University
February 12, 2019
We will describe recent progress on the existence theory and asymptotic analysis for solutions of the complex Ginzburg-Landau equations on closed manifolds, emphasizing connections to the existence of weak minimal submanifolds of codimension two. On manifolds with nontrivial first cohomology group, our results rely on new estimates for the Ginzburg-Landau energies along paths of maps connecting distinct homotopy classes of circle-valued maps, which may be of independent interest.

On the topology and index of minimal surfaces

Davi Maximo
University of Pennsylvania; Member, School of Mathematics
February 5, 2019
For an immersed minimal surface in $R^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resolves several conjectures made by J. Choe and D.

Spacetime positive mass theorem

Lan-Hsuan Huang
University of Connecticut; von Neumann Fellow, School of Mathematics
February 5, 2019
It is fundamental to understand a manifold with positive scalar curvature and its topology. The minimal surface approach pioneered by R. Schoen and S.T. Yau have advanced our understanding of positively curved manifolds. A very important result is their resolution to the Riemannian positive mass theorem. In general relativity, the concepts of positive scalar curvature and minimal surfaces naturally extend. The extensions connect to a more general statement, so-called the spacetime positive mass conjecture.