Yujie Wu (Stanford)
Firstly, we apply the method of generalized soap bubbles (µ-bubbles) to study manifolds with positive scalar curvature; we 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, including a 1-Urysohn width bound and bandwidth estimate for such 3-manifolds. Lastly, the method of θ-bubble allows us to swap the assumption of positive scalar curvature when using the µ-bubble method with the assumption of positive mean curvature of the boundary, obtaining analogous rigidity results for free boundary minimal hypersurfaces.
