Paper · Sep 24, 2018

The continuity equation, Hermitian metrics and elliptic bundles

Morgan Sherman, Ben Weinkove

math.DGarXiv:1809.08710
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Abstract

We extend the continuity equation of La Nave-Tian to Hermitian metrics and establish its interval of maximal existence. The equation is closely related to the Chern-Ricci flow, and we illustrate this in the case of elliptic bundles over a curve of genus at least two.

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Submitted
Sep 24, 2018
Subjects
math.DG
Journal
J. Geom. Anal. 30 (2020), no. 1, 762--776

Comments: 14 pages

Ricci flowGeneralized & G₂ flows

Its neighbourhood

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  1. The Chern-Ricci flow

    Valentino Tosatti and Ben Weinkove

    Topics Generalized & G₂ flows · Same author Ben Weinkove · Shared terms “Hermitian metrics”, “Chern Ricci”

  2. On the Calabi estimate of geometric flows of Hermitian metrics

    Marco Gallo and Luigi Vezzoni

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

  3. Hermitian metrics on complex non-Kähler manifolds

    Daniele Angella

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

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

    Lucio Bedulli and Luigi Vezzoni

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

  5. Geometric formalities along the Chern-Ricci flow

    Daniele Angella and Tommaso Sferruzza

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

  6. On Gauduchon connections with Kähler-like curvature

    Daniele Angella, Antonio Otal, Luis Ugarte et al.

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

  7. Chern-Ricci flow on Kato surfaces

    Daniele Angella and Mauricio Corrêa

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

  8. Regularity of solutions to Monge–Ampère equations on compact Hermitian manifolds

    Quang-Tuan Dang

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

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)

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.