## Ricci flows with Rough Initial Data

Peter Topping

University of Warwick

March 8, 2019

Ernst Kuwert

University of Freiburg

March 7, 2019

Abstract: For surfaces immersed into a compact Riemannian manifold, we consider the curvature functional given by the $L^2$ integral of the second fundamental form. We discuss an an area bound in terms of that functional, with application to the existence of minimizers (joint work with V. Bangert).

Stéphane Sabourau

Université Paris-Est Créteil

March 7, 2019

Gerard Besson

Université de Grenoble

March 7, 2019

Abstract : It is a joint work with G. Courtois, S. Gallot and A.Sambusetti. We prove a compactness theorem for metric spaces with anupper bound on the entropy and other conditions that will be discussed.Several finiteness results will be drawn. It is a work in progress.

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.

Neshan Wickramasekera

University of Cambridge; Member, School of Mathematics

March 6, 2019

Nancy Hingston

The College of New Jersey

March 6, 2019

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.

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.

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.