Paper · Nov 30, 2016

Schauder estimates for equations with cone metrics, I

Bin Guo, Jian Song

math.DGarXiv:1612.00075
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As on arXiv

Abstract

This is the first paper in a series to develop a linear and nonlinear theory for elliptic and parabolic equations on Kähler varieties with mild singularities. Donaldson has established a Schauder estimate for linear and complex Monge-Ampère equations when the background Kähler metrics on have cone singularities along a smooth complex hypersurface. We prove a sharp pointwise Schauder estimate for linear elliptic and parabolic equations on with background metric for . Our results give an effective elliptic Schauder estimate of Donaldson and a direct proof for the short time existence of the conical Kähler-Ricci flow.

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Submitted
Nov 30, 2016
Subjects
math.DG
Ricci flowKähler–Ricci flowSingularitiesKähler geometryHarmonic maps & DeTurck trick

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Connected papers

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 Compactness of Kahler-Ricci solitons on Fano manifolds (Bin Guo, Duong H. Phong, Jian Song et al., 2018)3Guo 2018 Schauder estimates for equations with cone metrics, II (Bin Guo and Jian Song, 2018)1Guo 2018 Kahler-Ricci flow on blowups along submanifolds (Bin Guo, 2018)5Guo 2018 Diameter estimates in Kähler geometry (Bin Guo, Duong H. Phong, Jian Song et al., 2022)2Guo 2022 The L^(∞) estimate for parabolic complex Monge-Ampère equations (Qizhi Zhao, 2023)7Zhao 2023 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)4Hallgren 2023 A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds (Wangjian Jian, Jian Song and Gang Tian, 2023)8Jian 2023 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)6Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Schauder estimates for equations with cone metrics, II

    Bin Guo and Jian Song

    Topics Harmonic maps & DeTurck trick, Kähler–Ricci flow, Kähler geometry · Same authors Bin Guo, Jian Song · Shared terms “Schauder estimate”, “cone metrics”, “singularities along”

  2. Diameter estimates in Kähler geometry

    Bin Guo, Duong H. Phong, Jian Song et al.

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same authors Bin Guo, Jian Song · Shared terms “Ampère equation”, “Kähler metrics”, “Monge Ampère”

  3. Compactness of Kahler-Ricci solitons on Fano manifolds

    Bin Guo, Duong H. Phong, Jian Song et al.

    Topics Kähler geometry, Singularities · Same authors Bin Guo, Jian Song

  4. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al.

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Jian Song

  5. Kahler-Ricci flow on blowups along submanifolds

    Bin Guo

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Bin Guo

  6. Type I estimates for the Kahler-Ricci flow I

    Wangjian Jian and Jian Song

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Jian Song

  7. The estimate for parabolic complex Monge-Ampère equations

    Qizhi Zhao

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “complex Monge”, “Ampère equation”, “Monge Ampère”

  8. A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds

    Wangjian Jian, Jian Song and Gang Tian

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Jian Song

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) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.