Geometry

I will prove Gromov's conjecture that every 3-manifold of positive scalar curvature contains a short closed geodesic. The proof uses Min-Max theory of minimal surfaces and a combinatorial version of mean curvature flow. Time permitting, I will describe other results about geometry and…

Llarull proved that the round n-sphere is extremal, meaning that one cannot simultaneously increase both its scalar curvature and its metric. Goette and Semmelmann generalized this result to spin maps f:M→N of nonzero Â-degree onto certain Riemannian manifolds with nonnegative curvature operator…

I'll discuss joint work with Adrian Chu, relating Kapouleas's doubling construction for minimal surfaces to the variational theory for a Coulomb-type interaction energy for Schroedinger operators. Namely, for the Jacobi operator of a given nondegenerate minimal surface, we show that families of…

In the past decades, we have witnessed rapid development in the construction of minimal surfaces with controlled topology by Simon-Smith min-max theory. In this talk, I'll discuss the existence of a number of genus 2 minimal surfaces in a 3-sphere with a positive-Ricci-curved metric. This…

The special Lagrangian equation (SLE) is a fully nonlinear elliptic PDE that originates in the work of Harvey and Lawson on calibrated geometries. The question whether a viscosity solution to the SLE is smooth (or at least has minimal gradient graph) is delicate, and the answer depends on the…

It is known from general relativity that axisymmetric stationary black holes can be reduced to axisymmetric harmonic maps into the hyperbolic plane H^2, while in the Riemannian setting, 4d Ricci-flat metrics with torus symmetry can also be locally reduced to such harmonic maps satisfying a…

Starting from the celebrated results of Eells and Sampson, a rich and flourishing literature has developed around equivariant harmonic maps from the universal cover of Riemann surfaces into nonpositively curved target spaces. In particular, such maps are known to be rigid, in the sense that they…

I will discuss the existence and regularity of critical points of the area functional among Lagrangian surfaces in symplectic 4-manifolds, reviewing classical results by R. Schoen and J. Wolfson, as well as recent progress by A. Pigati and T. Rivière. I will then present a variational…

We show that the Gromov simplicial volume of closed nonpositively curved 4-manifolds with negative Ricci curvature is positive, which is a partial resolution of Gromov's long-standing conjecture. This is done by giving a lower bound in terms of the Euler number and a generalized Gauss-Bonnet…

A key problem in geometric measure theory is to understand singularities in minimal surfaces when multiplicity occurs in the tangent cone. For questions concerning partial regularity, the primary situation to understand is that of a branch point, namely a (non-immersed) singular point where one…