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About
President
Governance
Partner Institutions
Visit
People
Management
Faculty
Postdocs
Visiting Scholars
Staff
Research
Research Groups
Courses
Seminars
Join Us
Faculty
Postdocs
Students
Events
Conferences
Workshops
Forum
Life @ BIMSA
Accommodation
Transportation
Facilities
Tour
News
News
Announcement
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Qiuzhen College, Tsinghua University
Yau Mathematical Sciences Center, Tsinghua University (YMSC)
Tsinghua Sanya International  Mathematics Forum (TSIMF)
Shanghai Institute for Mathematics and  Interdisciplinary Sciences (SIMIS)
BIMSA > BIMSA General Relativity Seminar Optimal Regularity and Uhlenbeck Compactness in Lorentzian Geometry and Beyond
Optimal Regularity and Uhlenbeck Compactness in Lorentzian Geometry and Beyond
Organizers
Lars Andersson , Bo Wen Zhao
Speaker
Moritz Reintjes
Time
Monday, May 22, 2023 4:00 PM - 5:00 PM
Venue
A3-3-301
Online
Zoom 928 682 9093 (BIMSA)
Abstract
We prove that curvature alone controls the derivatives of a connection, including the gravitational metrics of General Relativity, and the Yang-Mills connections of Particle Physics. Specifically, we prove that the regularity of $L^p$ connections can be lifted by coordinate/gauge transformation to one derivative above their $L^p$ bounded Riemann curvature, (i.e., to optimal regularity), thereby removing apparent singularities in the underlying geometry. This extends the classical result of Kazdan-DeTurck for Riemannian metric connections. As an application to General Relativity, our optimal regularity result implies that the Lipschitz continuous metrics of shock wave solutions of the Einstein-Euler equations are non-singular, (geodesic curves, locally inertial coordinates and the Newtonian limit all exist in a classical sense). By the extra connection derivative, we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry, and from compact to non-compact gauge groups. The proofs are based on our discovery of, and existence theory for, a novel system of non-linear partial differential equations, (the RT-equations), non-invariant equations which are elliptic independent of metric signature, and which provide a general procedure for constructing coordinate and gauge transformations that regularize spacetime connections.
Beijing Institute of Mathematical Sciences and Applications
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