Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare Conjecture

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A01=Qi S. Zhang
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ancient
Author_Qi S. Zhang
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Backward Limits
Category1=Non-Fiction
Category=PBK
Category=PBM
Compact Riemann Manifold
Complete Riemann Manifold
Conjugate Point
COP=United States
Covariant Derivative
curvature
Curvature Tensor
Delivery_Pre-order
eq_isMigrated=2
Euclidean geometry
Exponential Map
Hamilton’s little loop conjecture
harnack
Harnack Inequality
Heat Kernel Estimate
heat kernel estimates
inequality
Injectivity Radius
Jacobi Field
Language_English
log
Log Sobolev Inequality
manifold
manifolds
Nash Inequality
Ordinary Differential Equation
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Perelman’s entropies
Poincaré conjecture
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Ricci Curvature
Ricci Flow
riemann
Riemann Geometry
Riemann Manifold
Riemannian geometry
scalar
Scalar Curvature
Smooth Vector Fields
Sobolev imbedding
Sobolev inequalities
Sobolev Inequality
softlaunch
Surgery Cap
surgery theory
tensor
Tensor Field
V2 Ln V2
Vector Field

Product details

  • ISBN 9781439834596
  • Weight: 780g
  • Dimensions: 156 x 234mm
  • Publication Date: 02 Jul 2010
  • Publisher: Taylor & Francis Inc
  • Publication City/Country: US
  • Product Form: Hardback
  • Language: English
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Focusing on Sobolev inequalities and their applications to analysis on manifolds and Ricci flow, Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture introduces the field of analysis on Riemann manifolds and uses the tools of Sobolev imbedding and heat kernel estimates to study Ricci flows, especially with surgeries. The author explains key ideas, difficult proofs, and important applications in a succinct, accessible, and unified manner.

The book first discusses Sobolev inequalities in various settings, including the Euclidean case, the Riemannian case, and the Ricci flow case. It then explores several applications and ramifications, such as heat kernel estimates, Perelman’s W entropies and Sobolev inequality with surgeries, and the proof of Hamilton’s little loop conjecture with surgeries. Using these tools, the author presents a unified approach to the Poincaré conjecture that clarifies and simplifies Perelman’s original proof.

Since Perelman solved the Poincaré conjecture, the area of Ricci flow with surgery has attracted a great deal of attention in the mathematical research community. Along with coverage of Riemann manifolds, this book shows how to employ Sobolev imbedding and heat kernel estimates to examine Ricci flow with surgery.

Qi S. Zhang is a professor of mathematics at the University of California, Riverside.