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3-circle Theorem for Willmore surfaces II--degeneration of the complex structure

Yuxiang Li, Hao Yin, Jie Zhou

Abstract

We study the compactness of Willmore surfaces without assuming the convergence of the induced complex structures. In particular, we compute the energy loss in the neck in terms of the residue and we prove that the limit of the image of the Gauss map is a geodesic in the Grassmannian $G(2,n)$ whose length can also be computed in terms of the residue. Moreover, we provide a family of explicit Willmore surfaces in $\R^3$ that illustrate the denegeration phenomenon involved in the above results.

3-circle Theorem for Willmore surfaces II--degeneration of the complex structure

Abstract

We study the compactness of Willmore surfaces without assuming the convergence of the induced complex structures. In particular, we compute the energy loss in the neck in terms of the residue and we prove that the limit of the image of the Gauss map is a geodesic in the Grassmannian whose length can also be computed in terms of the residue. Moreover, we provide a family of explicit Willmore surfaces in that illustrate the denegeration phenomenon involved in the above results.

Paper Structure

This paper contains 25 sections, 30 theorems, 390 equations.

Table of Contents

  1. Introduction
  2. 3-circle lemma for elliptic equation
  3. A 3-circle lemma for linear elliptic equations
  4. 3-circle lemmas for nonlinear equations
  5. Asymptotic limit of the neck
  6. 3-circle lemma for the second fundamental forms
  7. Proof of Theorem \ref{['main1']}
  8. Proof of Theorem \ref{['neck.first.order']}
  9. Higher order expansions of the neck
  10. Proof of Theorem \ref{['main2']}
  11. Proof of Theorem \ref{['2nd.asymp.']}
  12. A corollary of Theorem \ref{['2nd.asymp.']}
  13. Proof of Theorem \ref{['thm:main']}
  14. Easy case.
  15. General case: energy and length
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $f_k$ be a conformal and Willmore immersion which satisfies A1)-A3). Then Moreover, the limit of the image of the Gauss map is a geodesic in $G(2,n)$ of length

Theorems & Definitions (62)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 52 more