Page 352 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 352
GEOMETRIC ANALYSIS AND LOW-DIMENSIONAL TOPOLOGY (MS-59)
A Local Singularity Analysis for the Ricci Flow
Reto Buzano, reto.buzano@unito.it
Queen Mary University of London and Università degli Studi di Torino, Italy
Coauthor: Gianmichele Di Matteo
The Ricci Flow is the most famous and most successful geometric flow, having led to resolutions
of the Poincaré and Geometrisation Conjectures, as well as proofs of the Differentiable Sphere
Theorem and the Generalised Smale Conjecture. For many of these applications, it is important
to understand precisely how singularities form along the flow - which is a notoriously difficult
task, in particular in dimensions strictly greater than three. In this talk, we develop a new and
refined singularity analysis for the Ricci Flow by investigating curvature blow-up rates locally.
We introduce general definitions of Type I and Type II singular points and show that these are
indeed the only possible types of singular points in a Ricci Flow. In particular, near any singular
point the Riemannian curvature tensor has to blow up at least at a Type I rate, generalising a
result previously obtained with Enders and Topping under a global Type I assumption. We also
prove analogous results for the Ricci tensor, as well as a localised version of Sesum’s result,
namely that the Ricci curvature must blow up near every singular point of a Ricci flow, again at
least at a Type I rate. If time permits, we will also see some applications of the theory to Ricci
flows with bounded scalar curvature.
Spaces of constrained positive scalar curvature metrics
Alessandro Carlotto, alessandro.carlotto@math.ethz.ch
ETH, Switzerland
In this lecture, I will present a collection of results concerning the interplay between the scalar
curvature of a Riemannian manifold and the mean curvature of its boundary, with special
emphasis on dimension-dependent phenomena. Our work is motivated by a network of far-
reaching conjectures by Gromov on the one hand, and by the study of the space of admissible
initial data sets for the Einstein field equation in general relativity on the other.
Foliation of Asymptotically Schwarzschild Manifolds by Generalized
Willmore Surfaces
Alexander Friedrich, alexfrie@math.ku.dk
University of Copenhagen, Denmark
From the perspective of general relativity asymptotically Schwarzschild, or more generally
asymptotically flat, manifolds represent isolated systems. Here the idea is that in the absents of
classical energy the spacetime should resemble the Minkowski space. The Hawing energy is a
quasi local energy of general relativity that strong relation to the Willmore functional. It was
introduced by S.W. Hawking in order to measure the gravitational energy of spacetimes that are
classically empty.
Starting from the Hawking energy we develop a notion of generalized Willmore func-
tionals. Further, we construct a foliation of the asymptotically flat end of an asymptotically
Schwarzschild manifold by large, area constrained spheres which are critical with respect to a
350
A Local Singularity Analysis for the Ricci Flow
Reto Buzano, reto.buzano@unito.it
Queen Mary University of London and Università degli Studi di Torino, Italy
Coauthor: Gianmichele Di Matteo
The Ricci Flow is the most famous and most successful geometric flow, having led to resolutions
of the Poincaré and Geometrisation Conjectures, as well as proofs of the Differentiable Sphere
Theorem and the Generalised Smale Conjecture. For many of these applications, it is important
to understand precisely how singularities form along the flow - which is a notoriously difficult
task, in particular in dimensions strictly greater than three. In this talk, we develop a new and
refined singularity analysis for the Ricci Flow by investigating curvature blow-up rates locally.
We introduce general definitions of Type I and Type II singular points and show that these are
indeed the only possible types of singular points in a Ricci Flow. In particular, near any singular
point the Riemannian curvature tensor has to blow up at least at a Type I rate, generalising a
result previously obtained with Enders and Topping under a global Type I assumption. We also
prove analogous results for the Ricci tensor, as well as a localised version of Sesum’s result,
namely that the Ricci curvature must blow up near every singular point of a Ricci flow, again at
least at a Type I rate. If time permits, we will also see some applications of the theory to Ricci
flows with bounded scalar curvature.
Spaces of constrained positive scalar curvature metrics
Alessandro Carlotto, alessandro.carlotto@math.ethz.ch
ETH, Switzerland
In this lecture, I will present a collection of results concerning the interplay between the scalar
curvature of a Riemannian manifold and the mean curvature of its boundary, with special
emphasis on dimension-dependent phenomena. Our work is motivated by a network of far-
reaching conjectures by Gromov on the one hand, and by the study of the space of admissible
initial data sets for the Einstein field equation in general relativity on the other.
Foliation of Asymptotically Schwarzschild Manifolds by Generalized
Willmore Surfaces
Alexander Friedrich, alexfrie@math.ku.dk
University of Copenhagen, Denmark
From the perspective of general relativity asymptotically Schwarzschild, or more generally
asymptotically flat, manifolds represent isolated systems. Here the idea is that in the absents of
classical energy the spacetime should resemble the Minkowski space. The Hawing energy is a
quasi local energy of general relativity that strong relation to the Willmore functional. It was
introduced by S.W. Hawking in order to measure the gravitational energy of spacetimes that are
classically empty.
Starting from the Hawking energy we develop a notion of generalized Willmore func-
tionals. Further, we construct a foliation of the asymptotically flat end of an asymptotically
Schwarzschild manifold by large, area constrained spheres which are critical with respect to a
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