Paper · Jan 7, 2025

Prescribed Chern scalar curvature flow on compact Hermitian manifolds with negative Gauduchon degree

Weike Yu

math.DGWider flowsarXiv:2501.03570
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Abstract

In this paper, we present a unified flow approach to prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree. When the conformal class of its Hermitian metric contains a balanced metric, we give some sufficient conditions on the candidate curvature function which guaranties the convergence of the flow to a conformal Hermitian metric whose Chern scalar curvature is .

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Submitted
Jan 7, 2025
Subjects
math.DG

Comments: 12pages

Other curvature flowsScalar curvature

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  1. A note on the Chern-Yamabe flow on compact Hermitian manifolds with negative Gauduchon degree

    Weike YuWider flows

    Topics Scalar curvature · Same author Weike Yu · Shared terms “Gauduchon degree”, “negative Gauduchon”, “compact Hermitian”

  2. Continuation and convergence for the higher dimensional conformal Chern–Calabi flow

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    Topics Scalar curvature, Other curvature flows · Shared terms “Chern scalar”, “compact Hermitian”, “Hermitian manifolds”

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    Topics Scalar curvature · Shared terms “Chern scalar”, “Hermitian manifolds”

  4. Calabi flow with bounded scalar curvature (II)

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    Topics Scalar curvature, Other curvature flows

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

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    Shared terms “compact Hermitian”, “Hermitian manifolds”, “Hermitian metric”

  6. Calabi flow with bounded scalar curvature

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    Topics Scalar curvature, Other curvature flows

  7. On global Chern-Calabi flows on compact complex surfaces

    Li Ma and Li ShengWider flows

    Topics Other curvature flows · Shared terms “Chern”, “Gauduchon”, “conformal Hermitian”

  8. A topological gap theorem for the -systole of positive scalar curvature 3-manifolds

    Kai XuWider flows

    Topics Scalar curvature, Other curvature flows

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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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.