School of Mathematics

Normalized harmonic map flow

Michael Struwe
ETH Zürich
March 6, 2019

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 3-sphere to any closed N is zero; moreover, the 
harmonic map heat flow may blow up in finite time, and even the identity map from the 3-sphere to itself is not stable under this flow. 

Existence and uniqueness of Green's function to a nonlinear Yamabe problem

Yanyan Li
Rutgers University
March 6, 2019

Abstract: For a given finite subset S of a compact Riemannian manifold (M; g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and
sufficient condition for the existence and uniqueness of a conformal metric on $M \setminus S$ such that each point of S corresponds to an asymptotically flat end and
that the Schouten tensor of the new conformal metric belongs to the boundary of the given cone.  This is a joint work with Luc Nguyen.

Liouville Equations and Functional Determinants

Andrea Malchiodo
Scuola Normale Superiore
March 5, 2019

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 amounts to solving Liouville equations with principal parts of different order but all scaling invariant. We discuss some existence, uniqueness, non-uniqueness results and some open problems. This is joint work with M.Gursky and P.Esposito. 

 

Spacetime positive mass theorem

Lan-Hsuan Huang
University of Connecticut; von Neumann Fellow, School of Mathematics
March 5, 2019

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 backgrounds and recent progress toward this statement. 

Rellich Kondrachov Theorem for L^2 curvatures in arbitrary dimension- Tristan Rivière

Tristan Rivière
ETH Zürich; Member, School of Mathematics
March 5, 2019

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 analysis challenges.