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Neckpinch Dynamics for Asymmetric Surfaces Evolving by Mean Curvature Flow
 
Zhou Gang :  California Institute of Technology, Pasadena, California, USA
Dan Knopf :  University of Texas at Austin, Austin, Texas, USA
Israel Michael Sigal :  University of Toronto, Toronto, Ontario, Canada
Neckpinch Dynamics for Asymmetric Surfaces Evolving by Mean Curvature Flow
eBook ISBN:  978-1-4704-4415-0
Product Code:  MEMO/253/1210.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $46.80
Add to cart
Neckpinch Dynamics for Asymmetric Surfaces Evolving by Mean Curvature Flow
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Neckpinch Dynamics for Asymmetric Surfaces Evolving by Mean Curvature Flow
Zhou Gang :  California Institute of Technology, Pasadena, California, USA
Dan Knopf :  University of Texas at Austin, Austin, Texas, USA
Israel Michael Sigal :  University of Toronto, Toronto, Ontario, Canada
eBook ISBN:  978-1-4704-4415-0
Product Code:  MEMO/253/1210.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $46.80
Add to cart
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 253; 2018; 78 pp
    MSC: Primary 53; 35

    The authors study noncompact surfaces evolving by mean curvature flow (mcf). For an open set of initial data that are \(C^3\)-close to round, but without assuming rotational symmetry or positive mean curvature, the authors show that mcf solutions become singular in finite time by forming neckpinches, and they obtain detailed asymptotics of that singularity formation. The results show in a precise way that mcf solutions become asymptotically rotationally symmetric near a neckpinch singularity.
  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. The first bootstrap machine
    • 3. Estimates of first-order derivatives
    • 4. Decay estimates in the inner region
    • 5. Estimates in the outer region
    • 6. The second bootstrap machine
    • 7. Evolution equations for the decomposition
    • 8. Estimates to control the parameters $a$ and $b$
    • 9. Estimates to control the fluctuation $\phi $
    • 10. Proof of the Main Theorem
    • A. Mean curvature flow of normal graphs
    • B. Interpolation estimates
    • C. A parabolic maximum principle for noncompact domains
    • D. Estimates of higher-order derivatives
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Memoirs of the American Mathematical Society
Volume: 253; 2018; 78 pp
MSC: Primary 53; 35

The authors study noncompact surfaces evolving by mean curvature flow (mcf). For an open set of initial data that are \(C^3\)-close to round, but without assuming rotational symmetry or positive mean curvature, the authors show that mcf solutions become singular in finite time by forming neckpinches, and they obtain detailed asymptotics of that singularity formation. The results show in a precise way that mcf solutions become asymptotically rotationally symmetric near a neckpinch singularity.
  • Chapters
  • 1. Introduction
  • 2. The first bootstrap machine
  • 3. Estimates of first-order derivatives
  • 4. Decay estimates in the inner region
  • 5. Estimates in the outer region
  • 6. The second bootstrap machine
  • 7. Evolution equations for the decomposition
  • 8. Estimates to control the parameters $a$ and $b$
  • 9. Estimates to control the fluctuation $\phi $
  • 10. Proof of the Main Theorem
  • A. Mean curvature flow of normal graphs
  • B. Interpolation estimates
  • C. A parabolic maximum principle for noncompact domains
  • D. Estimates of higher-order derivatives
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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