Geometry & Topology

The monotonicity formula is foundational to modern minimal surface theory. In $\mathbb R^3$, it says that any minimal surface has area growth at least that of a flat plane. I will first describe some new monotonicity formulas for minimal submanifolds of nonpositively curved symmetric spaces. I will then discuss applications to a program initiated by Gromov to prove statements of the following kind: Suppose we are given two manifolds $X$ and $Y$, where $X$ is “complicated” and $Y$ is lower dimensional. Then any map $f: X \to Y$ must have at least one “complicated” fiber. These include the first examples of higher expander families of Riemannian manifolds which gives a positive answer to a question of Gromov in the Riemannian setting. Finally if time permits I will discuss some applications to the topology of infinite volume locally symmetric spaces and/or the qualitative resolution of a conjecture of Farb on global fixed points for actions of higher rank lattices on low-dimensional contractible CAT(0) simplicial complexes. The latter two applications are based on a recent breakthrough of Connell–McReynolds–Wang. The talk should be accessible to a general audience and I will take care to explain necessary background on minimal surface theory and locally symmetric spaces. Joint work with Mikołaj Frączyk.