Stability of Einstein Manifolds
Author: Klaus Kröncke
Publisher:
Published: 2014-08-19
Total Pages: 156
ISBN-13: 9783838139081
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Stability of Asymptotically Hyperbolic Einstein Manifolds
Author: Yucheng Lu
Publisher:
Published: 2018
Total Pages: 62
ISBN-13: 9780438534711
DOWNLOAD EBOOKIn 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.
Essays on Einstein Manifolds
Author: Claude LeBrun
Publisher: American Mathematical Society(RI)
Published: 1999
Total Pages: 450
ISBN-13:
DOWNLOAD EBOOKThis is the sixth volume in a series providing surveys of differential geometry. It addresses: Einstein manifolds with zero Ricci curvature; rigidity and compactness of Einstein metrics; general relativity; the stability of Minkowski space-time; and more.
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
DOWNLOAD EBOOKV ? 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
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
DOWNLOAD EBOOKIn 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.
Stability and Hermitian-Einstein Metrics for Vector Bundles on Framed Manifolds
Author: Matthias Stemmler
Publisher:
Published: 2009
Total Pages: 0
ISBN-13:
DOWNLOAD EBOOKEinstein Manifolds
Author: Arthur L. Besse
Publisher: Springer Science & Business Media
Published: 2007-12-03
Total Pages: 529
ISBN-13: 3540741208
DOWNLOAD EBOOKEinstein's equations stem from General Relativity. In the context of Riemannian manifolds, an independent mathematical theory has developed around them. This is the first book which presents an overview of several striking results ensuing from the examination of Einstein’s equations in the context of Riemannian manifolds. Parts of the text can be used as an introduction to modern Riemannian geometry through topics like homogeneous spaces, submersions, or Riemannian functionals.
Complex Manifolds
Author: Steven Bell
Publisher: Springer Science & Business Media
Published: 1997-12-11
Total Pages: 324
ISBN-13: 9783540629955
DOWNLOAD EBOOKThe articles in this volume were written to commemorate Reinhold Remmert's 60th birthday in June, 1990. They are surveys, meant to facilitate access to some of the many aspects of the theory of complex manifolds, and demonstrate the interplay between complex analysis and many other branches of mathematics, algebraic geometry, differential topology, representations of Lie groups, and mathematical physics being only the most obvious of these branches. Each of these articles should serve not only to describe the particular circle of ideas in complex analysis with which it deals but also as a guide to the many mathematical ideas related to its theme.
Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics
Author: Y.-T. Siu
Publisher: Birkhäuser
Published: 2012-12-06
Total Pages: 172
ISBN-13: 3034874863
DOWNLOAD EBOOKThese notes are based on the lectures I delivered at the German Mathematical Society Seminar in Schloss Michkeln in DUsseldorf in June. 1986 on Hermitian-Einstein metrics for stable bundles and Kahler-Einstein metrics. The purpose of these notes is to present to the reader the state-of-the-art results in the simplest and the most comprehensible form using (at least from my own subjective viewpoint) the most natural approach. The presentation in these notes is reasonably self-contained and prerequisi tes are kept to a minimum. Most steps in the estimates are reduced as much as possible to the most basic procedures such as integration by parts and the maximum principle. When less basic procedures are used such as the Sobolev and Calderon-Zygmund inequalities and the interior Schauder estimates. references are given for the reader to look them up. A considerable amount of heuristic and intuitive discussions are included to explain why certain steps are used or certain notions introduced. The inclusion of such discussions makes the style of the presentation at some places more conversational than what is usually expected of rigorous mathemtical prese"ntations. For the problems of Hermi tian-Einstein metrics for stable bundles and Kahler-Einstein metrics one can use either the continuity method or the heat equation method. These two methods are so very intimately related that in many cases the relationship betwen them borders on equivalence. What counts most is the a. priori estimates. The kind of scaffolding one hangs the a.
Stability and Hermitian-Einstein Metrics for Vector Bundles on Framed Manifolds
Author:
Publisher:
Published: 2009
Total Pages:
ISBN-13:
DOWNLOAD EBOOKThe notions of stability of holomorphic vector bundles in the sense of Mumford-Takemoto and Hermitian-Einstein metrics in holomorphic vector bundles are adapted for canonically polarized framed manifolds, i. e. compact complex manifolds together with a smooth divisor admitting a certain projective embedding. The main tool is the Poincaré metric, a special complete Kähler-Einstein metric on the complement of the divisor, whose asymptotic behaviour near the divisor is well-known due to results by Schumacher. The existence and uniqueness of Hermitian-Einstein connections in stable holomorphic vector bundles (Kobayashi-Hitchin correspondence) is proved in the setting of framed manifolds.
