Seminar

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The avoidance principle says that two mean curvature flows of hypersurfaces remain disjoint if they are disjoint initially. In this talk, we will discuss several generalizations of the avoidance principle that allow for intersections between the two hypersurfaces. First, we show that the Hausdorff dimension of the intersection of two mean curvature flows is non-increasing over time. We then prove a dimension monotonicity result for self-intersections of immersed mean curvature flows. Next, we extend the intersection dimension monotonicity to a class of Brakke flows and level set flows, and we provide examples showing that this monotonicity fails for general Brakke flows. In the course of the proof, we obtain a localization result and an equivalence between non-fattening and non-discrepancy for level set flows with finitely many singularities. This is joint work with Tang-Kai Lee.