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A gluing construction of $D_{k}$ ALF gravitational instantons and existence of non-holomorphic minimal spheres

Xuwen Zhu

Abstract

This note extends the construction of $D_{k}$ ALF gravitational instantons in Schroers--Singer to a new case where the nonlinear superposition is given by the $D_{1}$ Atiyah--Hitchin metric and $k-1$ copies of $A_{0}$ Taub-NUT metrics. We then give a general class of ALF spaces such that each of them contains a non-holomorphic minimal sphere. Together with Foscolo's construction this gives a large class of $K3$ surfaces containing non-holomorphic minimal spheres.

A gluing construction of $D_{k}$ ALF gravitational instantons and existence of non-holomorphic minimal spheres

Abstract

This note extends the construction of ALF gravitational instantons in Schroers--Singer to a new case where the nonlinear superposition is given by the Atiyah--Hitchin metric and copies of Taub-NUT metrics. We then give a general class of ALF spaces such that each of them contains a non-holomorphic minimal sphere. Together with Foscolo's construction this gives a large class of surfaces containing non-holomorphic minimal spheres.
Paper Structure (7 sections, 7 theorems, 33 equations, 1 figure)

This paper contains 7 sections, 7 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The gluing construction of $D_{k}$ ALF spaces
  3. The multi-Taub-NUT space $M_{\epsilon}$
  4. The branched cover $\widehat{AH}_{\epsilon}$
  5. Gluing construction
  6. Solving for the hyperKähler triples
  7. Non-holomorphic minimal spheres

Key Result

Theorem 1

Given any $k\geq 2$ and any set of $k$ distinct points in $\mathbb R^{3}$ denoted as $\{0, p_{1}, \dots, p_{k-1}\}$ , there exists $\epsilon_{0}>0$, such that for any $\epsilon\in (0,\epsilon_{0})$ there exists a $D_{k}$ ALF metric $g_{\epsilon}$ which is a nonlinear superposition of the Atiyah--Hit

Figures (1)

  • Figure 1: An illustration of the resolved total space $\mathcal{W}$ obtained from gluing $\mathcal{W}_{0}$ and $\mathcal{W}_{1}$, with boundary faces and associated boundary defining functions on $\pi^{-1}(0)$.

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2: same as Theorem \ref{['thm2']}
  • Corollary 1: same as Corollary \ref{['cor1']}
  • Definition 1
  • Definition 2
  • Theorem 3
  • proof
  • Corollary 2
  • Remark
  • Theorem 4
  • ...and 5 more