# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### A conformally invariant gap theorem in Yang-Mills theory

We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem. This is joint work with Jeffrey Streets (UC Irvine) and Matthew Gursky (University of Notre Dame).

##### Almost Rigidity of the Positive Mass Theorem

The Positive Mass Theorem states that an asymptotically flat Riemannian manifold, $M^3$, with nonegative Scalar curvature has nonnegative ADM mass, $m_{ADM}(M)\ge 0$, and if the ADM mass is 0 then we have rigidity: the manifold is isometric to Euclidean space. It has long been known that if one has a sequence of such manifolds $M^3_j$ with $m_{ADM}(M_j) \to 0$ then $M_j$ need not converge smoothly to Euclidean space. To avoid bubbling, one forbids the sequence of manifolds to have closed interior minimal surfaces, but allows the manifolds to have minimal boundaries. Dan Lee and I conjectured that in this setting the $M_j$ converge to Euclidean space in the pointed intrinsic flat sense for well chosen points. We proved this in the rotationally symmetric setting and provided examples in that setting where smooth and Gromov-Hausdorff

##### Min-max theory for constant mean curvature hypersurfaces

We describe the construction of closed constant mean curvature (CMC) hypersurfaces using min-max methods. In particular, our theory allows us to show the existence of closed CMC hypersurfaces of any prescribed mean curvature in any closed Riemannian manifold. This work is joint with Xin Zhou.

##### Compactification of the configuration space for constant curvature conical metrics

In this joint work with Rafe Mazzeo, we would like to understand the deformation theory of constant curvature metrics with prescribed conical singularities on a compact Riemann surface. We construct a resolution of the configuration space, and prove a new regularity result that the family of constant curvature conical metrics has a nice compactification as the cone points coalesce. This is one key ingredient to understand the full moduli space of such metrics with positive curvature and cone angles bigger than $2\pi$.