Mean Curvature and Variational Theory

Explore recent developments in the existence theory of prescribing mean curvature (PMC) surfaces in this 45-minute lecture by Xin Zhou for the International Mathematical Union. Delve into mathematical models of soap bubbles and capillary surfaces, examining the class of PMC equations that describe them. Investigate minimal surfaces and constant mean curvature (CMC) surfaces as special cases of PMC surfaces. Learn about the variational theory perspective, where PMC surfaces are viewed as stationary points of the area functional plus a volume-related term. Survey key advancements, including the min-max theory for CMC/PMC surfaces and a Morse theory for the area functional. Gain insight into the recent solution of the Multiplicity One Conjecture for minimal surfaces using PMC min-max theory. Access accompanying slides for visual support and deeper understanding of the presented concepts.