Stationary $p$-harmonic maps approaching planar singular harmonic maps to the circle

TL;DR

This work addresses approximating singular harmonic maps into the circle $S^1$ by stationary $p$-harmonic maps in a planar domain as $p$ approaches $2$. It develops a two-step approach: first, constructing weak $p$-harmonic maps with prescribed singularities via phase perturbations of a fixed singular harmonic map, and second, relating the stationarity condition to the finite-dimensional criticality of the renormalised energy $W$ by showing the associated Dirac-coefficients converge to $ abla W$ as $p\uparrow 2$. The main result proves that from any topologically nondegenerate critical point of $W$ one can obtain stationary $p$-harmonic approximants for all $p$ near $2$, with singularities converging to the nondegenerate configuration and degrees carried by each singularity. This builds a bridge between the renormalised-energy geometry and variational limits of $p$-harmonic maps into $S^1$, clarifying how topology constrains stationarity and enabling constructive approximation of singular harmonic maps by $p$-harmonic critical points.

Abstract

Given a bounded planar domain $Ω\subset \mathbb{R}^2$, we show that any singular harmonic map into the circle $\mathbb{S}^1$ corresponding to a topologically nondegenerate critical point of the renormalised energy in the sense of Bethuel, Brezis and Hélein is a limit of stationary $p$-harmonic maps for $p < 2$ as $p \to 2$

Theorems & Definitions (32)

• Proposition 1.1
• Theorem 1
• Definition 1
• Proposition 2.1
• proof : Proof of Proposition \ref{['thm: singular HM']}
• Proposition 2.2
• proof
• Proposition 2.3
• proof : Proof of Proposition \ref{['prop: weak pharm']}
• Lemma 3.1