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Compactness of harmonic maps of surfaces with regular nodes

Woongbae Park

Abstract

In this paper, we formulate and prove a general compactness theorem for harmonic maps using Deligne-Mumford moduli space and families of curves. The main theorem shows that given a sequence of harmonic maps over a sequence of complex curves, there is a family of curves and a subsequence such that both the domains and the maps converge off the set of "non-regular" nodes. This provides a sufficient condition for a neck having zero energy and zero length. As a corollary, the following known fact can be proved: If all domains are diffeomorphic to $S^2$, both energy identity and zero distance bubbling hold.

Compactness of harmonic maps of surfaces with regular nodes

Abstract

In this paper, we formulate and prove a general compactness theorem for harmonic maps using Deligne-Mumford moduli space and families of curves. The main theorem shows that given a sequence of harmonic maps over a sequence of complex curves, there is a family of curves and a subsequence such that both the domains and the maps converge off the set of "non-regular" nodes. This provides a sufficient condition for a neck having zero energy and zero length. As a corollary, the following known fact can be proved: If all domains are diffeomorphic to , both energy identity and zero distance bubbling hold.

Paper Structure

This paper contains 12 sections, 23 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Harmonic Maps
  4. Families of Curves
  5. Convergence of Maps
  6. Neck analysis
  7. Continuity at Regular nodes
  8. Adding marked points to build new family
  9. Completion of the proof - Induction
  10. Appendix - Marking points lemmas
  11. Local properties
  12. Family properties

Key Result

Theorem \oldthetheorem

Suppose $f_{k} : C_{k} \rightarrow X$ be a sequence of harmonic maps with uniformly bounded energy defined on smooth $(g,n)$ curves. Then there is a subsequence $n_{k}$ and a way of marking points $P_{k}$ on $C_{k}$ such that corresponding sequence $f_{n_{k}} : C'_{n_{k}} = (C_{n_{k}},P_{n_{k}}) \ri

Theorems & Definitions (56)

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  • Corollary \oldthetheorem
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  • ...and 46 more