Paper · Jan 18, 2017

Weak Solutions of the Chern-Ricci flow on compact complex surfaces

Xiaolan Nie

math.DGarXiv:1701.04965
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Abstract

In this note, we prove the existence of weak solutions of the Chern-Ricci flow through blow downs of exceptional curves, as well as backwards smooth convergence away from the exceptional curves on compact complex surfaces. The smoothing property for the Chern-Ricci flow is also obtained on compact Hermitian manifolds of dimension n under a mild assumption.

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Submitted
Jan 18, 2017
Subjects
math.DG

Comments: 21 pages

Ricci flowGeneralized & G₂ flowsWeak & singular flows

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

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  1. The Chern-Ricci flow

    Valentino Tosatti and Ben Weinkove

    Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “Hermitian”

  2. The behavior of the second Ricci flow on complex parallelizable manifolds

    Lucio Bedulli and Luigi Vezzoni

    Topics Generalized & G₂ flows · Shared terms “compact complex”, “Chern Ricci”, “compact Hermitian”

  3. Geometric formalities along the Chern-Ricci flow

    Daniele Angella and Tommaso Sferruzza

    Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “Hermitian”

  4. Hermitian metrics on complex non-Kähler manifolds

    Daniele Angella

    Topics Generalized & G₂ flows · Shared terms “complex surfaces”, “compact complex”, “Chern Ricci”

  5. Regularizing properties of Complex Monge-Ampère flows II: Hermitian manifolds

    Tat Dat Tô

    Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “Hermitian manifolds”, “compact Hermitian”

  6. Hermitian manifolds with quasi-negative curvature

    Man-Chun Lee

    Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “Hermitian manifolds”, “compact Hermitian”

  7. An almost complex Chern-Ricci flow

    Tao Zheng

    Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “Hermitian”

  8. Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow

    Shaochuang Huang and Zhuo Peng

    Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “complex surfaces”, “Hermitian”

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.

  3. 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.