Monday

We describe a regularity result for equivariant harmonic maps from the universal cover of a Riemannian manifold into a (not necessarily locally finite) Euclidean building. As an application we prove non-Archimedean superrigidity for rank 1 symmetric spaces. This result extends the work of Gromov-Schoen, who proved p-adic superrigidity by considering locally finite targets. This work is joint with B. Dees and C. Mese.

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 are unique up to natural equivalence. Unfortunately, this rigidity property fails when the target space has positive curvature, and comparatively little is known in this framework.

In this talk, given a closed Riemann surface with strictly negative Euler characteristic and a unitary representation of its fundamental group on a separable complex Hilbert space H which is weakly equivalent to the regular representation, we aim to discuss a lower bound on the Dirichlet energy of equivariant harmonic maps from the universal cover of the surface into the unit sphere S of H, and to give a complete classification of the cases in which the equality is achieved. As a remarkable corollary, we obtain a lower bound on the area of equivariant minimal surfaces in S, and we determine all the representations for which there exists an equivariant, area-minimizing minimal surface in S.

The subject matter of this talk is a joint work with Antoine Song (Caltech) and Xingzhe Li (Cornell University).

Coassociatives are four-dimensional calibrated submanifolds of $G_2$ manifolds, seven dimensional manifolds with holonomy $G_2$. There is especially rich geometry to be studied when two coassociatives intersect transversely inside the ambient $G_2$ manifold. Non-compact examples of this phenomenon involving the Harvey-Lawson $Sp(1)$ invariant coassociatives in $R^7$ have been studied by Lotay and Kapouleas, who use the $U(1)$ symmetry in these examples to show the transverse intersection can be resolved by gluing in a family of Lawlor necks. The topology of the resulting coassociative is that of a generalized connected sum: the connected sum of the original two coassociatives along their intersection circles. In this talk, I will report on progress towards a generalization of this gluing construction for compact coassociative submanifolds intersecting transversely in an arbitrary $G_2$ manifold. If time permits, I will also describe an invariant for certain types of such transverse coassociative pairs.

We show that the entropy of a submanifold can be bounded from below by its (normalized) conformal volume. A consequence is the partial confirmation of a conjecture of Colding-Minicozzi about the entropy of higher codimension closed submanifolds.

We will discuss a classification of all compact and non compact convex ancient free boundary curve shortening flows in any convex domain of the Euclidean plane.