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The generalised energy identity and length of necks for $\varepsilon$-harmonic maps

Andrew M. Roberts

TL;DR

The paper addresses energy distribution and neck formation in the bubbling limits of $\varepsilon$-harmonic maps, extending Li–Wang’s generalized energy identity to a fourth-order perturbation. It develops a blow-up framework using stress-energy tensors and Pohozaev-type identities to decompose the limit into a smooth map, bubbles, and a neck contribution weighted by $\mu_i=\lim_{k\to\infty}\frac{\varepsilon_k}{r_{i,k}^2}\log(1/r_{i,k})$, with an intrinsic criterion that the energy identity holds if and only if $\mu_i\to1$ for all bubbles. The neck-length analysis introduces $\nu=\lim_k\sqrt{\frac{\varepsilon_k}{r_k^2}}\log(1/r_k)$, yielding no neck for $\nu=0$, a finite geodesic neck for $\nu\in(1,\infty)$ with length given by $\nu\sqrt{\frac{e^{-2\rho(0)}}{\pi}\int_{\mathbb{R}^2}|\Delta\omega|^2dx}$, or infinite neck for $\nu=\infty$, and it notes the possibility of energy-identity failure via neck formation. Altogether, the work extends the EI framework to $\varepsilon$-energy and clarifies energy transfer during bubbling through explicit geometric and analytic quantities.

Abstract

In this paper we find analogues for $\varepsilon$-harmonic maps to the generalised energy identity and the existence of geodesic necks result discovered by Yuxiang Li and Youde Wang for $α$-harmonic maps. In particular there exist specific quantities depending only on $\varepsilon$ and the bubbling radius which entirely determine if the full energy identity holds and if a neck forms. In the case these fail we can calculate the energy lost and the length of the geodesic neck based on only these quantities and the biharmonic energy of the bubble.

The generalised energy identity and length of necks for $\varepsilon$-harmonic maps

TL;DR

The paper addresses energy distribution and neck formation in the bubbling limits of -harmonic maps, extending Li–Wang’s generalized energy identity to a fourth-order perturbation. It develops a blow-up framework using stress-energy tensors and Pohozaev-type identities to decompose the limit into a smooth map, bubbles, and a neck contribution weighted by , with an intrinsic criterion that the energy identity holds if and only if for all bubbles. The neck-length analysis introduces , yielding no neck for , a finite geodesic neck for with length given by , or infinite neck for , and it notes the possibility of energy-identity failure via neck formation. Altogether, the work extends the EI framework to -energy and clarifies energy transfer during bubbling through explicit geometric and analytic quantities.

Abstract

In this paper we find analogues for -harmonic maps to the generalised energy identity and the existence of geodesic necks result discovered by Yuxiang Li and Youde Wang for -harmonic maps. In particular there exist specific quantities depending only on and the bubbling radius which entirely determine if the full energy identity holds and if a neck forms. In the case these fail we can calculate the energy lost and the length of the geodesic neck based on only these quantities and the biharmonic energy of the bubble.
Paper Structure (3 sections, 7 theorems, 48 equations)

This paper contains 3 sections, 7 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Proof of the energy identity
  3. Proof of the neck result

Key Result

Theorem 1.1

Let $u_k\in C^\infty(M,N)$ be a sequence of $\alpha_k$-harmonic maps with $0<\alpha_k\rightarrow1$ and $E_{\alpha_k}[u_k]$ uniformly bounded. Further, assume that there exists precisely one point of energy concentration $x\in M$. Now fix some local isothermal coordinates in a neighbourhood $U$ of $x with $F(x)=0$ and $F_*g=e^{2\rho(x)}(dx_1^2+dx_2^2)$ for some smooth $\rho$. Then there exist $u_\i

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof : Proof of Theorem \ref{['theorem: generalised EI epsilon']}
  • proof : Proof of Theorem \ref{['theorem: generalised EI dirichlet alpha']}
  • proof : Proof Theorem \ref{['theorem: neck result']}