Special Year 2018-19: Variational Methods in Geometry

Abstract: Let M be a compact Riemannian manifold. Morse theory for the energy function on the free loopspace LM of M gives a link between geometry and topology, between the growth of the index of the iterates of closed geodesics on M, and the...

Abstract: Finding non-constant harmonic 3-spheres for a closed target manifold N is a prototype of a super-critical variational problem. In fact, the
direct method fails, as the infimum of the Dirichlet energy in any homotopy class of maps from the...

Abstract: Functional Determinants are quantities constructed out of spectra of conformally covariant operators, and are explicit in dimension two and four, due to formulas by Polyakov and Branson-Oersted. Extremizing them in a conformal class...

Abstract: I will talk about periodic geodesics, geodesic loops, and geodesic nets on Riemannian manifolds. More specifically, I will discuss some curvature-free upper bounds for compact manifolds and the existence results for non-compact manifolds...

Abstract: The spacetime positive mass theorem says that an asymptotically flat initial data set with the dominant energy condition must have a timelike energy-momentum vector, unless the initial data set is in the Minkowski spacetime. We will review...

Abstract : What are the possible limits of smooth curvatures with uniformly bounded $L^p$ norms ?We shall see that the attempts to give a satisfying answer to this natural question from the calculus of variation of gauge theory brings us to numerous...

Abstract: After recalling the gluing construction for Kaehler constant scalar curvature and extremal (`a la Calabi) metrics starting from a compact or ALE orbifolds with isolated singularities, I will show how to compute the Futaki invariant of the...

Abstract: Colding-Minicozzi introduced a natural entropy for hypersurfaces in euclidean space that is non-increasing under the mean curvature flow (MCF) and is a natural measure of the hypersurface's geometric complexity. In particular...