abstracts trends-2024

We show that there exists a quantity, depending only on C^0 data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the C^0 sense and has nonnegative scalar curvature in the sense of Ricci flow. Moreover, the C^0 mass at infinity is independent of choice of $C^0$-asymptotically flat coordinate chart, and the C^0 local mass has controlled distortion under Ricci-DeTurck flow when coupled with a suitably evolving test function.

Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes manifolds with pointwise quarter-pinched sectional curvatures and manifolds with positive curvature operator. The PIC condition is preserved under the Ricci flow, and the construction of a flow with surgeries was used to give a complete classification of PIC manifolds in dimension 4 and in sufficiently high dimensions. There is a completely different approach to the study of PIC manifolds using minimal surfaces. In this talk we will give an overview of these two approaches, and discuss recent results on largeness properties of stable surfaces in PIC manifolds.

By the compactness theory of Bamler, any finite time singularity of the Kahler-Ricci flow is modeled on a singular Kahler-Ricci soliton, and such solitons are infinitesimally metric cones. In this talk, we will see that these cones are normal affine algebraic varieties, using a new method for proving Hormander-type L^2 estimates on singular shrinking solitons.

We will discuss a new approach to flows with surgery, which in particular applies in the setting of mean-convex surfaces with free boundary. As an application (joint with Dan Ketover) we establish the existence of multiple free boundary minima