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Examples of Toric Scalar-flat Kähler Surfaces with Mixed-type Ends

Yueqing Feng

Abstract

Given a strictly unbounded toric symplectic 4-manifold, we explicitly construct complete toric scalar-flat Kähler metrics on the complement of a toric divisor. These symplectic 4-manifolds correspond to a specific class of non-compact Kähler surfaces. We also provide an alternative construction of toric scalar-flat Kähler metrics with conical singularity along the toric divisor, following the approach of Abreu and Sena-Dias.

Examples of Toric Scalar-flat Kähler Surfaces with Mixed-type Ends

Abstract

Given a strictly unbounded toric symplectic 4-manifold, we explicitly construct complete toric scalar-flat Kähler metrics on the complement of a toric divisor. These symplectic 4-manifolds correspond to a specific class of non-compact Kähler surfaces. We also provide an alternative construction of toric scalar-flat Kähler metrics with conical singularity along the toric divisor, following the approach of Abreu and Sena-Dias.

Paper Structure

This paper contains 7 sections, 18 theorems, 141 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline of the article.
  3. Acknowledgements
  4. Preliminaries
  5. Construction of scalar-flat Kähler metrics of Poincaré type
  6. A conical family of toric metrics
  7. Appendix: Uniqueness of toric metrics under given boundary conditions

Key Result

Theorem 1.1

(Theorem Theorem unbounded PTK SFK+Theorem theorem asymptotic behavior) Given $X$ a strictly unbounded toric symplectic $4$-manifold and $D=\sum_{i=1}^mD_i$ a divisor on $X$ such that each $D_i$ is an irreducible prime divisor fixed by the torus action and $D_i\cap D_j=\emptyset$ for any $1\le i\ne

Figures (2)

  • Figure 1: The moment polytope $P\backslash\ell_I$
  • Figure 2: The moment polytope of the Hwang-Singer metric

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • Definition 2.3
  • Theorem 2.1
  • Definition 2.4
  • ...and 32 more