Regularity Theory for Mean Curvature Flow

Author: Klaus Ecker
Publisher: Springer Science & Business Media
ISBN: 0817682104
Format: PDF, Mobi
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* Devoted to the motion of surfaces for which the normal velocity at every point is given by the mean curvature at that point; this geometric heat flow process is called mean curvature flow. * Mean curvature flow and related geometric evolution equations are important tools in mathematics and mathematical physics.

Lecture Notes on Mean Curvature Flow Barriers and Singular Perturbations

Author: Giovanni Bellettini
Publisher: Springer
ISBN: 8876424296
Format: PDF, Docs
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The aim of the book is to study some aspects of geometric evolutions, such as mean curvature flow and anisotropic mean curvature flow of hypersurfaces. We analyze the origin of such flows and their geometric and variational nature. Some of the most important aspects of mean curvature flow are described, such as the comparison principle and its use in the definition of suitable weak solutions. The anisotropic evolutions, which can be considered as a generalization of mean curvature flow, are studied from the view point of Finsler geometry. Concerning singular perturbations, we discuss the convergence of the Allen–Cahn (or Ginsburg–Landau) type equations to (possibly anisotropic) mean curvature flow before the onset of singularities in the limit problem. We study such kinds of asymptotic problems also in the static case, showing convergence to prescribed curvature-type problems.

Nonlinear Partial Differential Equations

Author: Mi-Ho Giga
Publisher: Springer Science & Business Media
ISBN: 9780817646516
Format: PDF, ePub
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This work will serve as an excellent first course in modern analysis. The main focus is on showing how self-similar solutions are useful in studying the behavior of solutions of nonlinear partial differential equations, especially those of parabolic type. This textbook will be an excellent resource for self-study or classroom use.

Global Differential Geometry

Author: Christian Bär
Publisher: Springer Science & Business Media
ISBN: 3642228429
Format: PDF
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This volume contains a collection of well-written surveys provided by experts in Global Differential Geometry to give an overview over recent developments in Riemannian Geometry, Geometric Analysis and Symplectic Geometry. The papers are written for graduate students and researchers with a general interest in geometry, who want to get acquainted with the current trends in these central fields of modern mathematics.

Handbook of geometric analysis

Author: Lizhen Ji
Publisher: Intl Pr of Boston Inc
ISBN: 9781571461308
Format: PDF, Docs
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Geometric Analysis combines differential equations with differential geometry. An important aspect of geometric analysis is to approach geometric problems by studying differential equations. Besides some known linear differential operators such as the Laplace operator, many differential equations arising from differential geometry are nonlinear. A particularly important example is the Monge-Amperè equation. Applications to geometric problems have also motivated new methods and techniques in differential equations. The field of geometric analysis is broad and has had many striking applications.This handbook of geometric analysis—the first of the two to be published in the ALM series—presents introductions and survey papers treating important topics in geometric analysis, with their applications to related fields. It can be used as a reference by graduate students and by researchers in related areas.

Third International Congress of Chinese Mathematicians

Author: Ka-Sing Lau
Publisher: Amer Mathematical Society
ISBN:
Format: PDF, ePub, Mobi
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This volume consists of the proceedings of the Third International Congress of Chinese Mathematicians, held at the Chinese University of Hong Kong in December 2004. The Congress brought together eminent Chinese and overseas mathematicians to discuss the latest developments in pure and applied mathematics.

The Ricci Flow

Author:
Publisher: American Mathematical Soc.
ISBN:
Format: PDF, ePub, Mobi
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Entropy, $\mu$-invariant, and finite time singularities Geometric tools and point picking methods Geometric properties of $\kappa$-solutions Compactness of the space of $\kappa$-solutions Perelman's pseudolocality theorem Tools used in proof of pseudolocality Heat kernel for static metrics Heat kernel for evolving metrics Estimates of the heat equation for evolving metrics Bounds for the heat kernel for evolving metrics Elementary aspects of metric geometry Convex functions on Riemannian manifolds Asymptotic cones and Sharafutdinov retraction Solutions to selected exercises Bibliography Index