Paper · Aug 3, 2017

Minimizers of the sharp Log entropy on manifolds with non-negative Ricci curvature and flatness

Qi S Zhang

math.DGarXiv:1708.01049
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Abstract

Consider the scaling invariant, sharp log entropy (functional) introduced by Weissler on noncompact manifolds with nonnegative Ricci curvature. It can also be regarded as a sharpened version of Perelman's W entropy in the stationary case. We prove that it has a minimizer if and only if the manifold is isometric to . Using this result, it is proven that a class of noncompact manifolds with nonnegative Ricci curvature is isometric to . Comparing with the well known flatness results in, and on asymptotically flat manifolds and asymptotically locally Euclidean (ALE) manifolds, their decay or integral condition on the curvature tensor is replaced by the condition that the metric converges to the Euclidean one in sense at infinity. No second order condition on the metric is needed.

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Submitted
Aug 3, 2017
Subjects
math.DG

Comments: 10 pages

Perelman entropyBoundaries & non-compact

Its neighbourhood

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A rigidity result for ancient Ricci flows

    Qi S. Zhang

    Topics Perelman entropy · Same author Qi S Zhang · Shared terms “log entropy”, “sharp log”

  2. Extremals of Log Sobolev inequality on non-compact manifolds and Ricci soliton structures

    Michele Rimoldi and Giona Veronelli

    Topics Perelman entropy, Boundaries & non-compact

  3. Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds

    Zeraoulia Rafik and Sobhan Sobhan Allah

    Topics Perelman entropy · Shared terms “Comparing”

  4. On the singularity formation of gauge fields coupled to Ricci flow

    Andoni Royo Abrego

    Topics Perelman entropy · Shared terms “scaling invariant”

  5. Classical and quantum geometric information flows and entanglement of relativistic mechanical systems

    Sergiu I. Vacaru and Laurenţiu Bubuianu

    Topics Perelman entropy

  6. Stochastic Thermodynamics on Time-Evolving Curved Spaces

    Rihito Nagase, Shoki Sugimoto, Asuka Takatsu et al.

    Topics Perelman entropy

  7. On the geometry of asymptotically conical Ricci shrinkers (II): strong uniqueness

    Yu Li, Longhuan Yan and Wenjia Zhang

    Topics Perelman entropy, Boundaries & non-compact

  8. A local Sobolev inequality on Ricci flow and its applications

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Perelman entropy Perelman found that Ricci flow is a gradient flow once a measure is fixed. His 𝓕-functional and 𝓦-entropy increase along the flow; a lower bound… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6B.16
    Eigenvalue Problems and Perelman’s λBook 6B · Linear PDE: Characteristics, Waves and Regularity

    Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.