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Schauder estimates for equations with cone metrics, II

Bin Guo, Jian Song

Abstract

This is the continuation of our paper \cite{GS}, 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.

Schauder estimates for equations with cone metrics, II

Abstract

This is the continuation of our paper \cite{GS}, 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.

Paper Structure

This paper contains 28 sections, 70 theorems, 464 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Elliptic case.
  5. Parabolic case.
  6. Compact Kähler manifolds
  7. A useful lemma
  8. Elliptic estimates
  9. Conical harmonic functions
  10. Smooth approximating metrics
  11. Cheng-Yau gradient estimate revisit
  12. Laplacian estimate in singular directions
  13. Mixed derivatives estimates
  14. Convergence of $u_\epsilon$
  15. Tangential and Laplacian estimates
  16. ...and 13 more sections

Key Result

Theorem \oldthetheorem

Suppose ${\boldsymbol\beta}\in (1/2,1)^p$ and $f\in C^0(B_{\boldsymbol\beta}(0,1))$ is Dini continuous with respect to $g_{\boldsymbol\beta}$. Let $u\in C^0(\overline{B_{\boldsymbol\beta}(0,1)})\cap C^2(B_{\boldsymbol\beta}(0,1)\backslash \mathcal{S})$ be the solution to the equation eqn:main equati for any $1\le j\le p$, and for any $1\le j, k\le p$ with $j\neq k$, where $d=d_{\boldsymbol\beta}

Theorems & Definitions (141)

  • Definition 1.1
  • Theorem \oldthetheorem
  • Remark 1.1
  • Corollary 1.1
  • Remark 1.2
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary 1.2
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 131 more