Ricci Flow and the Sphere Theorem

Ricci Flow and the Sphere Theorem
Free download. Book file PDF easily for everyone and every device. You can download and read online Ricci Flow and the Sphere Theorem file PDF Book only if you are registered here. And also you can download or read online all Book PDF file that related with Ricci Flow and the Sphere Theorem book. Happy reading Ricci Flow and the Sphere Theorem Bookeveryone. Download file Free Book PDF Ricci Flow and the Sphere Theorem at Complete PDF Library. This Book have some digital formats such us :paperbook, ebook, kindle, epub, fb2 and another formats. Here is The CompletePDF Book Library. It's free to register here to get Book file PDF Ricci Flow and the Sphere Theorem Pocket Guide.

Contents:

  • Donate to arXiv
  • Categories
  • Ricci Flow and the Sphere Theorem
  • Account Options
  • [v2] Curvature, sphere theorems, and the Ricci flow

    • In Sect. These equations, particularly that of Theorem 4.

    Donate to arXiv

    An important foundational step in the study of any system of evolutionary partial differential equations is to show short-time existence and uniqueness. For the Ricci flow, unfortunately, short-time existence does not follow from standard parabolic theory, since the flow is only weakly parabolic. To overcome this, Hamilton's seminal paper [Ham82b] employed the deep Nash —Moser implicit function theorem to prove short-time existence and uni- queness.

    A detailed exposition of this result and its applications can be found in Hamilton's survey [Ham82a]. DeTurck [DeT83]later found a more direct proof by modifying the flow by a time-dependent change of variables to make it parabolic. It is this method that we will follow. In Theorem 4. The maximum principle is the main tool we will use to understand the behaviourof solutions to the Ricci flow. While other problems arising in geo- metric analysis and calculus of variations make strong use of techniques from functional analysis, here — due to the fact that the metric is changing — most of these techniques are not available; although methods in this direction are developed in the work of Perelman [Per02].

    Categories

    The maximum principle, though very simple, is also a very powerful tool which can be used to show that pointwise inequalities on the initial data of parabolic pde are preserved by the evolution. As we have already seen, when the metric evolves by Ricci flow the various curvature tensors R, Ric, and Scal do indeed satisfy systems of parabolic pde.

    Our main applications of the maximum principle will be to prove that certain inequalities on these tensors are preserved by the Ricci flow, so that the geometry of the evolving metrics is controlled. In Chaps.

    Ricci Flow and the Sphere Theorem

    By appealing to this view, we would expect the same kind of regularity that is seen in parabolic equa- tions to apply to the curvature. In particular we want to show that bounds on curvature automatically induce a priori bounds on all derivatives of the curvature for positive times. In the literature these are known as Bernstein— Bando—Shi derivative estimates as they follow the strategy and techniques introduced by Bernstein done in the early twentieth century for proving gradient bounds via the maximum principle and were derived for the Ricci flow in [Ban87] and comprehensively by Shi in [Shi89].

    In the second section we use these bounds to prove long-time existence. The compactness theorem for the Ricci flow tells us that any sequence of complete solutions to the Ricci flow, having uniformly bounded curvature and injectivity radii uniformly bounded from below, contains a convergent subsequence. This result has its roots in the convergence theory developed by Cheeger and Gromov.

    Ricci Flow and the Sphere Theorem

    In many contexts where the latter theory is applied, the regularity is a crucial issue. By contrast, the proof of the compactness theorem for the Ricci flow is greatly aided by the fact that a sequence of solutions to the Ricci flow enjoy excellent regularity properties which were discussed in the previous chapter. Indeed, it is precisely because bounds on the curvature of a solution to the Ricci flow imply bounds on all derivatives of the curvature that the compactness theorem produces C8 -convergence on compact sets.

    The limiting solution obtained from these gives information about the structure of the singularity. As a remark concerning notation in this chapter, quantities depending on the metric gk or gk t will have a subscript k. For instance?

    Kundrecensioner

    Quantities without a subscript will depend on the background metric g. After Ricci flow was first introduced, it appeared for many years that there was no variational characterisation of the flow as the gradient flow of a geometric quantity. View online Borrow Buy Freely available Show 0 more links Set up My libraries How do I set up "My libraries"?

    • Sphere theorem - Wikipedia.
    • Simon Brendle: “Ricci Flow and the Sphere Theorem”.
    • The Unfolded Protein Response and Cellular Stress, Part A.
    • The Ricci Flow in Riemannian Geometry.

    Australian National University Library. Open to the public. Monash University Library.

    Account Options

    Open to the public ; The University of Melbourne Library. Open to the public Book English University of Adelaide. Barr Smith Library. Open to the public ; Online : Licence restrictions may apply Book English Show 0 more libraries Open to the public Book English Show 0 more libraries None of your libraries hold this item. Found at these bookshops Searching - please wait We were unable to find this edition in any bookshop we are able to search. These online bookshops told us they have this item:.

    Tags What are tags? Add a tag.

    1. Outrageous Advertising Thats Outrageously Successful: Created for the 99% of Small Business Owners Who Are Dissatisfied with the Results They Get From Their Current Advertising.
    2. Commentary on the United Nations Convention on the Rights of the Child: Volume 24 Article 24: the Right to Health And Health Services (Commentary on the ... Convention on the Rights of the Child.).
    3. Sphere Theorems | Andreas Savas-Halilaj.
    4. Singularities of Differentiable Maps: Volume I: The Classification of Critical Points Caustics and Wave Fronts.
    5. Tragedys Spirit.
    6. Time and Place.
    7. Donate to arXiv.

    Public Private login e. Add a tag Cancel Be the first to add a tag for this edition.

    Poincaré Conjecture - Numberphile

    Lists What are lists? Login to add to list.

    [v2] Curvature, sphere theorems, and the Ricci flow

    Be the first to add this to a list. Comments and reviews What are comments?

    Add a comment. Australian National University.

    Monash University. Queensland University of Technology. The University of Melbourne.