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The Ricci Flow in Riemannian Geometry

A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem

Ben Andrews, Christopher Hopper (Autoren)

Buch | Softcover

XVIII, 302 Seiten

2010 | 2011
Springer Berlin (Verlag)
978-3-642-16285-5 (ISBN)

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This book focuses on Hamilton's Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for Riemannian manifolds, and Perelman's noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of Böhm and Wilking and Brendle and Schoen have led to a proof of the differentiable 1/4-pinching sphere theorem.

1 Introduction.- 2 Background Material.- 3 Harmonic Mappings.- 4 Evolution of the Curvature.- 5 Short-Time Existence.- 6 Uhlenbeck's Trick.- 7 The Weak Maximum Principle.- 8 Regularity and Long-Time Existence.- 9 The Compactness Theorem for Riemannian Manifolds.- 10 The F-Functional and Gradient Flows.- 11 The W-Functional and Local Noncollapsing.- 12 An Algebraic Identity for Curvature Operators.- 13 The Cone Construction of Böhm and Wilking.- 14 Preserving Positive Isotropic Curvature.- 15 The Final Argument

From the reviews:

"The book is dedicated almost entirely to the analysis of the Ricci flow, viewed first as a heat type equation hence its consequences, and later from the more recent developments due to Perelman's monotonicity formulas and the blow-up analysis of the flow which was made thus possible. ... is very enjoyable for specialists and non-specialists (of curvature flows) alike." (Alina Stancu, Zentralblatt MATH, Vol. 1214, 2011)

Erscheint lt. Verlag	25.11.2010
Reihe/Serie	Lecture Notes in Mathematics
Zusatzinfo	XVIII, 302 p. 13 illus., 2 illus. in color.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	484 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	35-XX, 53-XX, 58-XX • Partial differential equations • Ricci Flow • Riemannian Geometry • Sphere theorem
ISBN-10	3-642-16285-1 / 3642162851
ISBN-13	978-3-642-16285-5 / 9783642162855
Zustand	Neuware