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Explore a mathematical lecture examining the nonlocal version of the classical Hoffman-Meeks halfspace theorem for minimal surfaces. Learn about the fundamental 1990 result by Hoffman and Meeks, which established that any connected, proper, possibly branched minimal surface in three-dimensional Euclidean space contained within a halfspace must be a plane, and understand why this theorem fails to extend to higher dimensions due to counterexamples like the higher-dimensional catenoid. Discover the corresponding nonlocal problem concerning sets in Euclidean space with zero s-mean curvature that are contained in a halfspace, and examine whether such sets must themselves be halfspaces. Study the proof techniques demonstrating that under specific conditions - when the boundary of the set is continuously differentiable or when it satisfies universal volumetric density estimates at all scales - the set must indeed be a halfspace. Understand how this nonlocal version differs significantly from the classical case by holding true in all dimensions, not just three-dimensional space. The presentation covers joint research work with Matteo Cozzi and was delivered as part of the Thematic Programme on Free Boundary Problems at the Erwin Schrödinger International Institute for Mathematics and Physics.
