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Pengzi Miao : Remarks on a Scalar Curvature Rigidity Theorem of Brendle and Marques

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In a recent paper, Brendle and Marques proved that on certain geodesic balls in the standard hemisphere, there does not exist small metric deformations of the standard metric which increase the scalar curvature in the interior and the mean curvature on the boundary. Such a result was motivated by the Euclidean and Hyperbolic positive mass theorems. More interestingly, this result is false on the hemisphere itself, which is shown by Brendle-Marques-Neves' remarkable counter example to the Min-Oo's conjecture. In this talk, we provide a few remarks to Brendle and Marques' theorem. We show that their theorem remains valid on slightly larger geodesic balls; it also holds on certain convex domains; moreover, with a volume constraint imposed, a variation of their theorem holds on the hemisphere. This is a joint work with Luen-Fai Tam.

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