Paper · Sep 10, 2018

Schauder estimates for equations with cone metrics, II

Bin Guo, Jian Song

math.DGarXiv:1809.03116
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Abstract

This is the continuation of our paper, to study the linear theory for equations with conical singularities. We derive interior Schauder estimates for linear elliptic and parabolic equations with a background Kähler metric of conical singularities along a divisor of simple normal crossings. As an application, we prove the short-time existence of the conical Kähler-Ricci flow with conical singularities along a divisor with simple normal crossings.

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Submitted
Sep 10, 2018
Subjects
math.DG, math.AP
Journal
Analysis & PDE 17 (2024) 757-830
DOI
10.2140/apde.2024.17.757

Comments: Comments are welcome!

Ricci flowKähler–Ricci flowSingularitiesKähler geometryHarmonic maps & DeTurck trick

Its neighbourhood

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 Schauder estimates for equations with cone metrics, I (Bin Guo and Jian Song, 2016)1Guo 2016 Compactness of Kahler-Ricci solitons on Fano manifolds (Bin Guo, Duong H. Phong, Jian Song et al., 2018)5Guo 2018 Kahler-Ricci flow on blowups along submanifolds (Bin Guo, 2018)3Guo 2018 Diameter estimates in Kähler geometry (Bin Guo, Duong H. Phong, Jian Song et al., 2022)2Guo 2022 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)4Hallgren 2023 Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle (Jiawei Liu and Xi Zhang, 2024)6Liu 2024 Failure of the local decay formula for Arnold multiplicities under twisted Kähler–Ricci flow (Xiangsen Qin, 2026)8Qin 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)7Jian 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, I

    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

  3. Kahler-Ricci flow on blowups along submanifolds

    Bin Guo

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

  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. 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

  6. Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle

    Jiawei Liu and Xi Zhang

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “conical Kähler”, “singularities along”

  7. 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

  8. Failure of the local decay formula for Arnold multiplicities under twisted Kähler–Ricci flow

    Xiangsen Qin

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “normal crossings”, “simple normal”, “short time”

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.