Paper · Apr 18, 2018

Short-Time Existence Theorem for the CR Torsion Flow

Shu-Cheng Chang, Chin-Tung Wu

math.DGarXiv:1804.06585
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Abstract

In this paper, we study the torsion flow which is served as the CR analogue of the Ricci flow in a closed pseudohermitian manifold. We show that there exists a unique smooth solution to the CR torsion flow in a small time interval with the CR pluriharmonic function as an initial data. In spirit, it is the CR analogue of the Cauchy-Kovalevskaya local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems

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Submitted
Apr 18, 2018
Subjects
math.DG

Comments: 30 pages

Ricci flowStability & uniquenessHarmonic maps & DeTurck trick

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  1. A simple proof of the short-time existence and uniqueness for Ricci flow

    Kaveh Eftekharinasab

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”, “local existence”

  2. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  3. Ricci flow on Finsler surfaces

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    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  4. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  5. Short-time existence of the Ricci flow on complete, non-collapsed -manifolds with Ricci curvature bounded from below

    Raphael Hochard

    Topics Harmonic maps & DeTurck trick · Shared terms “short time”, “time existence”, “initial data”

  6. On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five

    Shu-Cheng Chang, Fengjiang Li, Chien Lin et al.

    Topics Ricci flow · Same authors Shu-Cheng Chang, Chin-Tung Wu · Shared terms “existence theorem”

  7. Geometry and Topology of Gradient Shrinking Sasaki-Ricci Solitons

    Shu-Cheng Chang, Yingbo Han and Chin-Tung Wu

    Same authors Shu-Cheng Chang, Chin-Tung Wu · Shared terms “analogue”

  8. Existence of the -Ricci-Yamabe flow on closed manifolds

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    Topics Harmonic maps & DeTurck trick · Shared terms “existence theorem”, “time existence”, “short time”

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.