Paper · Oct 17, 2017

Lie-Algebraic Curvature Conditions preserved by the Hermitian Curvature Flow

Yury Ustinovskiy

math.CVarXiv:1710.06035
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Abstract

The purpose of this paper is to prove that the Hermitian Curvature Flow (HCF) on an Hermitian manifold preserves many natural curvature positivity conditions. Following Wilking, for an -invariant subset and a ncie function we construct a convex set of curvature operators , which is invariant under the HCF. Varying and , we prove that the HCF preserves Griffiths positivity, Dual-Nakano positivity, positivity of holomorphic orthogonal bisectional curvature, lower bounds on the second scalar curvature. As an application, we prove that periodic solutions to the HCF can exist only on manifolds with the trivial canonical bundle on the universal cover .

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Submitted
Oct 17, 2017
Subjects
math.CV, math.DG
DOI
10.1007/s00208-020-01965-y

Comments: 20 pages, 1 figure

Generalized & G₂ flowsScalar curvature

Its neighbourhood

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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.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper The Hermitian curvature flow on manifolds with non-negative Griffiths curvature (Yury Ustinovskiy, 2016)1Ustinovskiy 2016 Hermitian curvature flow on complex homogeneous manifolds (Yury Ustinovskiy, 2017)3Ustinovskiy 2017 Hermitian Curvature flow on unimodular Lie groups and static invariant metrics (Ramiro A. Lafuente, Mattia Pujia and Luigi Vezzoni, 2018)8Lafuente 2018 Expanding solitons to the Hermitian curvature flow on complex Lie groups (Mattia Pujia, 2018)6Pujia 2018 Hermitian manifolds with quasi-negative curvature (Man-Chun Lee, 2018)5Lee 2018 Hermitian Curvature Flow on compact homogeneous spaces (Francesco Panelli and Fabio Podestà, 2019)7Panelli 2019 On the Structure of Hermitian Manifolds with Semipositive Griffiths Curvature (Yury Ustinovskiy, 2019)2Ustinovskiy 2019 Griffiths positivity for Bismut curvature and its behaviour along Hermitian Curvature Flows (Giuseppe Barbaro, 2021)4Barbaro 2021
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Hermitian curvature flow on manifolds with non-negative Griffiths curvature

    Yury Ustinovskiy

    Topics Generalized & G₂ flows · Same author Yury Ustinovskiy · Shared terms “Hermitian curvature”, “HCF”, “Griffiths”

  2. On the Structure of Hermitian Manifolds with Semipositive Griffiths Curvature

    Yury Ustinovskiy

    Topics Generalized & G₂ flows · Same author Yury Ustinovskiy · Shared terms “HCF”, “Hermitian curvature”, “Hermitian manifolds”

  3. Hermitian curvature flow on complex homogeneous manifolds

    Yury Ustinovskiy

    Topics Generalized & G₂ flows · Same author Yury Ustinovskiy · Shared terms “HCF”, “Hermitian curvature”, “Lie”

  4. Griffiths positivity for Bismut curvature and its behaviour along Hermitian Curvature Flows

    Giuseppe Barbaro

    Topics Generalized & G₂ flows · Shared terms “Hermitian curvature”, “Griffiths positivity”, “HCF”

  5. Hermitian manifolds with quasi-negative curvature

    Man-Chun Lee

    Topics Generalized & G₂ flows · Shared terms “Hermitian curvature”, “Hermitian manifolds”, “bisectional curvature”

  6. Expanding solitons to the Hermitian curvature flow on complex Lie groups

    Mattia Pujia

    Topics Generalized & G₂ flows · Shared terms “Hermitian curvature”, “HCF”, “algebraic”

  7. Hermitian Curvature Flow on compact homogeneous spaces

    Francesco Panelli and Fabio Podestà

    Topics Generalized & G₂ flows · Shared terms “Hermitian curvature”

  8. Hermitian Curvature flow on unimodular Lie groups and static invariant metrics

    Ramiro A. Lafuente, Mattia Pujia and Luigi Vezzoni

    Topics Generalized & G₂ flows · Shared terms “Hermitian curvature”, “HCF”, “Lie”

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.

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.