Schedule:

2:30 pm- 3:30 pm (First Talk)

3:30 pm- 4:00 pm Break

4:00 pm- 5:00 pm (Second Talk) 

 

Abstract:

In these two talks, I will discuss recent work with Christos Mantoulidis on the supremum of the total boundary mean curvature over compact 3-manifolds with nonnegative scalar curvature, with a prescribed boundary metric. When the manifold boundary consists of topological 2-spheres, we demonstrate that the finiteness of the supremum follows from the previous work of Wang-Yau and Shi-Tam on the quasi-local mass problem in general relativity. When the manifold boundary consists of arbitrary 2-surfaces, we establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. In the proof of the additivity property, we make use of the recent method of Mantoulidis-Schoen on computing the Bartnik mass of apparent horizons. A main feature of the results discussed here is that they do not require the Gauss curvature of the boundary to be positive. I plan to provide complete statement and background of these results in the first lecture and then provide more details of the proof in the second lecture.