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Scalar-flat Kähler metrics with varying cone angle singularities along a divisor

Gonçalo Oliveira, Rosa Sena-Dias

Abstract

Using an ansatz due to LeBrun we construct complete scalar-flat Kähler metrics with a prescribed varying conical singularity along a divisor.

Scalar-flat Kähler metrics with varying cone angle singularities along a divisor

Abstract

Using an ansatz due to LeBrun we construct complete scalar-flat Kähler metrics with a prescribed varying conical singularity along a divisor.
Paper Structure (6 sections, 7 theorems, 34 equations)

This paper contains 6 sections, 7 theorems, 34 equations.

Table of Contents

  1. Introduction and main result
  2. LeBrun's hyperbolic ansatz
  3. Proof of the main result
  4. The Dirichlet problem on hyperbolic space
  5. Local behaviour of the metric along the strict transform of $\mathbb{C} \times \lbrace 0 \rbrace$
  6. Asymptotic behaviour of the metrics and their completeness

Key Result

Theorem 1

Let $k \in \mathbb{N}_0$, $\xi_1, \ldots , \xi_k \in \mathbb{C} \times \lbrace 0 \rbrace \subset \mathbb{C}^2$ and $\mathrm{Bl}_{\xi_1, \ldots , \xi_k}\mathbb{C}^2$ be the blow-up of $\mathbb{C}^2$ at the points $\xi_1, \ldots , \xi_k$. Then, for any smooth positive function over $\mathbb{CP}^1 \tim

Theorems & Definitions (13)

  • Theorem 1
  • Remark 1
  • Theorem 2: LeBrun in LeBrun
  • Theorem 3: Andersson, Sullivan
  • Lemma 1
  • proof
  • Remark 2
  • Definition 1
  • Remark 3
  • Lemma 2
  • ...and 3 more