Stability of Asymptotically Hyperbolic Einstein Manifolds
Stability of Asymptotically Hyperbolic Einstein Manifolds

Author: Yucheng Lu

Publisher:

Published: 2018

Total Pages: 62

ISBN-13: 9780438534711

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In this thesis we study the stability of the Ricci flow. The stability problem of Ricci flow in different settings have been considered by Ye, Li-Yin, Schnurer-Schulze-Simon and Bamler etc. We consider a more general case and extend the results to the general case, that is, in the setting of asymptotically hyperbolic Einstein (AHE) manifolds with rough initial data. First we introduce the background of the problem and results on the long time behavior of Ricci flow in detail. Then we compare the difference in methodology of theses results and extend to the AHE case. We consider the normalized Ricci flow on a AHE manifold with initial metrics which are perturbations of a non-degenerate AHE metric h0. The key step is to obtain exponential decay of certain geometric quantities. Then we prove that the normalized Ricci flow converges exponentially fast to h0, if the perturbation is L2-bounded and C 0-small.

Stability by Linearization of Einstein's Field Equation
Stability by Linearization of Einstein's Field Equation

Author: Lluís Bruna

Publisher: Springer Science & Business Media

Published: 2013-11-25

Total Pages: 220

ISBN-13: 3034603045

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V ? V ?K? , 3 2 2 R ? /?x K i i g V T G g ?T , ? G g g 4 ? R ? ? G ? T g g ? h h ? 2 2 2 2 ? ? ? ? ? ? ? h ?S , ?? ?? 2 2 2 2 2 c ?t ?x ?x ?x 1 2 3 S T S T? T?. ? ̃ T S 2 2 2 2 ? ? ? ? ? ? ? h . ?? 2 2 2 2 2 c ?t ?x ?x ?x 1 2 3 g h h ?? g T T g vacuum M n R n R Acknowledgements n R Chapter I Pseudo-Riemannian Manifolds I.1 Connections M C n X M C M F M C X M F M connection covariant derivative M ? X M ×X M ?? X M X,Y ?? Y X ? Y ? Y ? Y X +X X X 1 2 1 2 ? Y Y ? Y ? Y X 1 2 X 1 X 2 ? Y f? Y f?F M fX X ? fY X f Y f? Y f?F M X X ? torsion ? Y?? X X,Y X,Y?X M . X Y localization principle Theorem I.1. Let X, Y, X , Y be C vector ?elds on M.Let U be an open set

Issues in Applied Mathematics: 2011 Edition
Issues in Applied Mathematics: 2011 Edition

Author:

Publisher: ScholarlyEditions

Published: 2012-01-09

Total Pages: 1316

ISBN-13: 1464965072

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Issues in Applied Mathematics / 2011 Edition is a ScholarlyEditions™ eBook that delivers timely, authoritative, and comprehensive information about Applied Mathematics. The editors have built Issues in Applied Mathematics: 2011 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about Applied Mathematics in this eBook to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in Applied Mathematics: 2011 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.

Emerging Topics on Differential Equations and Their Applications
Emerging Topics on Differential Equations and Their Applications

Author: Hua Chen

Publisher: World Scientific

Published: 2012

Total Pages: 319

ISBN-13: 981444975X

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The aim of the SinoOCoJapan Conference of Young Mathematicians was to provide a forum for presenting and discussing recent trends and developments in differential equations and their applications, as well as to promote scientific exchanges and collaborations among young mathematicians both from China and Japan.The topics discussed in this proceedings include mean curvature flows, KAM theory, N-body problems, flows on Riemannian manifolds, hyperbolic systems, vortices, water waves, and reaction diffusion systems.

Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds
Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds

Author: John M. Lee

Publisher: American Mathematical Soc.

Published: 2006

Total Pages: 83

ISBN-13: 9781470404680

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The main purpose of this monograph is to give an elementary and self-contained account of the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities sufficiently close to that of a given asymptotically hyperbolic Einstein metric with non positive curvature. The proof is based on an elementary derivation of sharp Fredholm theorems for self-adjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds.

General Relativity and the Einstein Equations
General Relativity and the Einstein Equations

Author: Yvonne Choquet-Bruhat

Publisher: Oxford University Press

Published: 2009

Total Pages: 812

ISBN-13: 0199230722

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General Relativity has passed all experimental and observational tests to model the motion of isolated bodies with strong gravitational fields, though the mathematical and numerical study of these motions is still in its infancy. It is believed that General Relativity models our cosmos, with a manifold of dimensions possibly greater than four and debatable topology opening a vast field of investigation for mathematicians and physicists alike. Remarkable conjectures have been proposed, many results have been obtained but many fundamental questions remain open. In this monograph, aimed at researchers in mathematics and physics, the author overviews the basic ideas in General Relativity, introduces the necessary mathematics and discusses some of the key open questions in the field.

Linear Stability of Einstein Metrics and Perelman's Lambda-functional for Manifolds with Conical Singularities
Linear Stability of Einstein Metrics and Perelman's Lambda-functional for Manifolds with Conical Singularities

Author: Changliang Wang

Publisher:

Published: 2016

Total Pages: 119

ISBN-13: 9781369146769

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In the second part, we prove that the spectrum of --4Delta + R consists of discrete eigenvalues with finite multiplicities on a compact Riemannian manifold of dimension n with a single conical singularity, if the scalar curvature of cross section of conical neighborhood is greater than n -- 2. Moreover, we obtain an asymptotic behavior for eigenfunctions near the singularity. As a consequence of these spectrum properties, we extend the theory of Perelman's lambda-functional on smooth compact manifolds to compact manifolds with isolated conical singularities.

AdS/CFT Correspondence
AdS/CFT Correspondence

Author: Olivier Biquard

Publisher: European Mathematical Society

Published: 2005

Total Pages: 264

ISBN-13: 9783037190135

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Since its discovery in 1997 by Maldacena, AdS/CFT correspondence has become one of the prime subjects of interest in string theory, as well as one of the main meeting points between theoretical physics and mathematics. On the physical side, it provides a duality between a theory of quantum gravity and a field theory. The mathematical counterpart is the relation between Einstein metrics and their conformal boundaries. The correspondence has been intensively studied, and a lot of progress emerged from the confrontation of viewpoints between mathematics and physics. Written by leading experts and directed at research mathematicians and theoretical physicists as well as graduate students, this volume gives an overview of this important area both in theoretical physics and in mathematics. It contains survey articles giving a broad overview of the subject and of the main questions, as well as more specialized articles providing new insight both on the Riemannian side and on the Lorentzian side of the theory.

Emerging Topics On Differential Equations And Their Applications - Proceedings On Sino-japan Conference Of Young Mathematicians
Emerging Topics On Differential Equations And Their Applications - Proceedings On Sino-japan Conference Of Young Mathematicians

Author: Yiming Long

Publisher: World Scientific

Published: 2012-12-31

Total Pages: 319

ISBN-13: 9814449768

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The aim of the Sino-Japan Conference of Young Mathematicians was to provide a forum for presenting and discussing recent trends and developments in differential equations and their applications, as well as to promote scientific exchanges and collaborations among young mathematicians both from China and Japan.The topics discussed in this proceedings include mean curvature flows, KAM theory, N-body problems, flows on Riemannian manifolds, hyperbolic systems, vortices, water waves, and reaction diffusion systems.