Talk Abstracts
Application of Curvature Comparison Theorems in Minimal Surfaces
- We use generalized minimal or capillary hypersurfaces to study comparison theorems of certain class of positively curved manifolds (with boundary); this is also called Gromov’s μ-bubble method. We apply these results to obtain useful geometric bounds such as Urysohn width, bandwidth estimates, and study rigidity of (free boundary) minimal hypersurfaces.
Capillary Hypersurfaces and Variational Methods
- We study free boundary and capillary minimal hypersurfaces from the variational point of view. In particular, we study the interaction of these objects with scalar curvature and boundary convexity. We first apply the method of free boundary µ-bubbles) to study manifolds with positive scalar curvature to prove a rigidity result for free boundary minimal hypersurfaces in a 4-manifolds with certain positivity assumptions on curvature. Then we define generalized capillary surfaces (θ-bubbles) and use θ-bubbles to obtain geometric estimates on manifolds with non-negative scalar curvature and uniformly mean convex boundary, obtaining estimates and rigidity results for such manifolds.
From μ-bubble to θ-bubble: Geometry of Mean Convex Manifolds with NNSC
- We introduce a method of constructing (generalized) capillary surfaces called “θ-bubble”, via Gromov’s celebrated “µ-bubble” method. Using this, we study manifolds with nonnegative scalar curvature (NNSC) and strictly mean convex boundary.
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