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The energy of maps accompanying the collapsing of the $K3$ surface to a flat 3-dimensional orbifold

Kota Hattori

Abstract

We study the Dirichlet energy of some smooth maps appearing in a collapsing family of hyper-Kähler metrics on the $K3$ surface constructed by Foscolo. We introduce an invariant for homotopy classes of smooth maps from the $K3$ surface with a hyper-Kähler metric to a flat Riemannian orbifold of dimension $3$, then show that it gives a lower bound of the energy. Moreover, we show that the ratio of the energy to the invariant converges to $1$ for Foscolo's collapsing families.

The energy of maps accompanying the collapsing of the $K3$ surface to a flat 3-dimensional orbifold

Abstract

We study the Dirichlet energy of some smooth maps appearing in a collapsing family of hyper-Kähler metrics on the surface constructed by Foscolo. We introduce an invariant for homotopy classes of smooth maps from the surface with a hyper-Kähler metric to a flat Riemannian orbifold of dimension , then show that it gives a lower bound of the energy. Moreover, we show that the ratio of the energy to the invariant converges to for Foscolo's collapsing families.

Paper Structure

This paper contains 13 sections, 22 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Hyper-Kähler triples
  3. Hyper-Kähler metrics on the $K3$ surface collapsing to $3$-dimensional spaces
  4. Gibbons-Hawking ansatz
  5. Hyper-Kähler metrics on the $K3$ surface
  6. Smooth maps
  7. Smooth maps from smooth manifold to $\mathbb{T}/\{ \pm 1\}$
  8. Smooth maps giving the collapsing to $\mathbb{T}/\{ \pm 1\}$
  9. Lower bounds of Dirichlet energy
  10. Linear maps from $\mathbb{R}^4$ to $\mathbb{R}^3$
  11. Estimates on the energy
  12. Estimates on Riemannian manifold
  13. Estimates on the energies and the integral invariants

Key Result

Theorem 1.1

Let $\varepsilon_0>0$ be a sufficiently small positive constant and $(g_\varepsilon)_{\varepsilon\in(0,\varepsilon_0]}$ be a family of hyper-Kähler metrics constructed in Foscolo2019 such that $\{ (X,g_\varepsilon)\}_\varepsilon$ converges to $(\mathbb{T}/\{ \pm 1\}, g_{\mathbb{T}})$ as $\varepsilon such that $\mathcal{E}(f)\ge \mathcal{I}_\varepsilon([f])$ for any $\varepsilon$ and any smooth map

Theorems & Definitions (53)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Lemma 2.8
  • Lemma 2.9
  • ...and 43 more