Asymptotic geometry of conformally overcomplete manifolds Dr. Achinta Kumar Nandi, University of California San Diego Host: Abdullah Al Helal Zoom talk
Abstract: The interior $(X,g)$ of a smooth $(n+1)$-dimensional Riemannian manifold $\bar{X}$ with boundary is called conformally overcomplete when $\rho^{\sigma} g$ is smooth up to the boundary, where $\sigma > 2$. We study the asymptotic behavior of geodesics in this setting. In particular, we show that limiting exponential map endows the boundary with an intrinsically determined smooth structure. As a consequence, it is established that isometries between conformally overcomplete manifolds extend to conformal diffeomorphisms of their boundaries and $\mathcal{C}^1$-diffeomorphisms of their closure. We also prove that the spectrum of the Laplace-Beltrami operator for conformally overcomplete manifolds is $[0,\infty)$. This is a work in progress with Dr. Sean Curry.