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s-stability for W^{s,n/s}-harmonic maps in homotopy groups

Katarzyna Mazowiecka, Armin Schikorra

Abstract

We study $s$-dependence for minimizing $W^{s,n/s}$-harmonic maps $u\colon \mathbb{S}^n \to \mathbb{S}^\ell$ in homotopy classes. Sacks--Uhlenbeck theory shows that, for each $s$, minimizers exist in a generating subset of $π_{n}(\mathbb{S}^\ell)$. We show that this generating subset can be chosen locally constant in $s$. We also show that as $s$ varies the minimal $W^{s,n/s}$-energy in each homotopy class changes continuously. In particular, we provide progress to a question raised by Mironescu and Brezis--Mironescu.

s-stability for W^{s,n/s}-harmonic maps in homotopy groups

Abstract

We study -dependence for minimizing -harmonic maps in homotopy classes. Sacks--Uhlenbeck theory shows that, for each , minimizers exist in a generating subset of . We show that this generating subset can be chosen locally constant in . We also show that as varies the minimal -energy in each homotopy class changes continuously. In particular, we provide progress to a question raised by Mironescu and Brezis--Mironescu.
Paper Structure (11 sections, 23 theorems, 149 equations)

This paper contains 11 sections, 23 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Continuity for the Sobolev-norm
  4. Existence of minimizers and energy identity
  5. Globally improved regularity in the conformal scaling
  6. Improved global estimates for harmonic maps
  7. A fractional version of Iwaniec's stability result
  8. Continuous dependence
  9. Continuous dependence of minimal energy: Proof of Theorem \ref{['th:continuousminenergy']}
  10. Proof of Corollaries
  11. Stability of generators of minimizing Wsns-harmonic maps: Proof of Theorem \ref{['th:generator2']}

Key Result

Theorem 1.1

For any $\ell,n \geq 1$, with either $(\ell,n) = (1,1)$ or $\ell \geq 2$, $s \in (0,1)$. There exists a generating set $X_s \subset \pi_{n}({\mathbb S}^\ell)$ such that for any $\alpha \in X_s$ the infimum $\#_s \alpha$ is attained.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark 1.7
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • ...and 31 more