There has recently been a great deal of activity on Scalar Curvature, both on the index theoretic and metric geometry side. The classic work of Gromov-Lawson and Schoen-Yau is a backdrop for the more modern developments, and here we will describe some recent work with Unger-Yau which seeks to generalize some of the classic results to the setting of non-compact manifolds. Connections with problems in Conformal Geometry, Positive Mass Theorems will be alluded to, and other current developments by other authors will be described. This is based on arXiv:2009.12618.
Geometric flows have been one of the most fruitful research areas in geometric analysis the past decades with vast applications such as the resolution of the Geometrization conjecture, the Riemannian Penrose Inequality and the differentiable sphere theorem. Often, the main goal here is to precisely understand the long-time behavior of these geometric flows. In 1996 Stahl studied free boundary mean curvature flow (i.e. mean curvature flow with Neumann boundary data) whose barrier surface is given by a round sphere and showed that convex surfaces converge to shrinking hemispheres along the flow. This can be compared to Huisken’s celebrated convergence result for mean curvature flow. An alternative approach to Stahl’s result has been given by Edelen in 2016. This raises the natural question whether the assumption of the barrier being a round sphere can be removed and that a similar result holds true for arbitrary convex barrier surfaces. Using a novel 5-tensor perturbation technique we answer this question affirmatively in the 2-dimensional setting. This work is joint with Martin Li.