Mini-courses:

  • Riemannian geometry in general relativity, by Dan Lee (CUNY).
    Abstract: I will explain how questions in general relativity lead to interesting problems in Riemannian geometry. This will involve a tour from the Einstein equations to the constraint equations, to the study of asymptotically flat initial data sets, ADM energy-momentum, the positive mass theorem, and the Penrose inequality.

  • Minimal surfaces and scalar curvature, by Otis Chodosh (Princeton/IAS).
    Abstract: I'll discuss the link between minimal surfaces and scalar curvature. For example, I will explain how minimal surfaces were used to prove that there is no metric on the 3-dimensional torus with positive scalar curvature by Schoen--Yau. I will also discuss the surprising link between such results and general relativity, as well as some related rigidity results. I'll assume basic knowledge of Riemannian geometry.

Invited Speakers:

  • Stability of graphical tori with almost nonnegative scalar curvature, by Raquel Perales.
    Abstract: By works of Schoen--Yau and Gromov--Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to $0$. We prove flat and intrinsic flat subconvergence to a flat torus for sequences of $3$-dimensional tori $M_j$ that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound, a uniform lower bound on the area of the smallest closed minimal surface, and scalar curvature bounds of the form $R_{g_{M_j}} \geq -1/j$. We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang--Lee, Huang--Lee--Sormani and Sormani.
    With A. Cabrera Pacheco and C. Ketterer.

Short Talks:

  • A uniqueness result for higher-dimensional Reissner--Nordström manifolds, by Sophia Jahns.
    Abstract: We consider n+1-dimensional static solutions of the electrovacuum equations which are asymptotic to a member of the Reissner--Nordström family, with a lapse and an electric potential fulfilling some asymptotic conditions. Assuming that we are given such a spacetime whose inner boundary (a priori possibly with multiple connected components) consists of static horizons or photon spheres (which are characterized by a quasilocal subextremality condition), we show that it is isometric to a subextremal Reissner--Nordström spacetime of positive mass. The proof relies on ideas going back to the well-known black hole uniquess thereom by Bunting and Masood-ul Alam and generalizes results by Cederbaum, Cederbaum--Galloway, and Lazov--Yazadjiev.