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On the blow-up of harmonic maps from surfaces to homogeneous manifolds

Hongcan Qian, Hao Yin

Abstract

We study harmonic map sequences from surfaces to compact homogeneous spaces. For sequences developing a single bubble, we derive refined asymptotic expansions in the neck region and prove new obstruction relations among the leading coefficients. These strengthen earlier results by converting an inequality into an equality. For weakly conformal maps, this yields geometric constraints: in low dimensions the tangent planes of the limit map and bubble must coincide, while in higher dimensions they are isoclinic.

On the blow-up of harmonic maps from surfaces to homogeneous manifolds

Abstract

We study harmonic map sequences from surfaces to compact homogeneous spaces. For sequences developing a single bubble, we derive refined asymptotic expansions in the neck region and prove new obstruction relations among the leading coefficients. These strengthen earlier results by converting an inequality into an equality. For weakly conformal maps, this yields geometric constraints: in low dimensions the tangent planes of the limit map and bubble must coincide, while in higher dimensions they are isoclinic.

Paper Structure

This paper contains 5 sections, 11 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Proof of the main theorem
  3. The normal part
  4. The tangent part
  5. Proof of Corollary \ref{['cor:main']}

Key Result

Theorem 1

Let $\{u_i\}$ be a sequence of harmonic maps from $B_1$ to $N$ with uniformly bounded energy, satisfying conditions (i)-(iii) as above. There exist uniformly bounded (vector) coefficients $p_i, q_i, a_{i}, b_{i}, c_{i}$ and $d_{i}$ for any $\alpha\in (0,1)$, such that $u_i$ regarded as function of t where $\eta=e^t+\lambda_ie^{-t}$ and $O(\eta^{1+\alpha})$ is some function $w$ satisfying Moreover

Theorems & Definitions (19)

  • Theorem 1
  • Remark 2
  • Corollary 3: Corollary 1.2 of yin
  • Theorem 4
  • Corollary 5
  • Lemma 6: Lemma 3.1 in yin
  • Lemma 7
  • Remark 8
  • proof
  • Lemma 9
  • ...and 9 more