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Morse Index Stability for Sequences of Sacks-Uhlenbeck Maps into a Sphere

Francesca Da Lio, Tristan Rivière, Dominik Schlagenhauf

TL;DR

This work analyzes sequences of $p_k$-harmonic maps into spheres with $p_k>2$ and uniformly bounded energy, establishing $L^2$ and $L^{2,1}$ energy quantization and a precise pointwise gradient control in neck regions. By showing necks do not contribute negative directions to the second variation, the authors prove an upper semicontinuity result for the extended Morse index plus nullity under bubble-tree convergence to a limit consisting of a base map and bubbles. The approach combines a refined Hodge decomposition, Lorentz-space estimates, Wente-type inequalities, entropy-type conditions, and spectral diagonalization with respect to neck weights, extending the DGR22 framework to the Sacks-Uhlenbeck setting for maps into $\mathbb{S}^n$ and yielding robust Morse-theoretic stability for conformally invariant 2D variational problems.

Abstract

In this paper we consider sequences of $p$-harmonic maps, $p>2$, from a closed Riemann surface $Σ$ into the $n$-dimensional sphere $\mathbb{S}^n$ with uniform bounded energy. These are critical points of the energy $E_p(u) :=\int_Σ\left( 1+|{\nabla u}|^2\right)^{p/2} \ dvol_Σ.$ Our two main results are an improved pointwise estimate of the gradient in the neck regions around blow up points and the proof that the necks are asymptotically not contributing to the negativity of the second variation of the energy $E_p.$ This allows us, in the spirit of the paper of the first and second authors in collaboration with M. Gianocca {\em Morse index stability for critical points to conformally invariant Lagrangians}, to show the upper semicontinuity of the Morse index plus nullity for sequences of $p$-harmonic maps into a sphere.

Morse Index Stability for Sequences of Sacks-Uhlenbeck Maps into a Sphere

TL;DR

This work analyzes sequences of -harmonic maps into spheres with and uniformly bounded energy, establishing and energy quantization and a precise pointwise gradient control in neck regions. By showing necks do not contribute negative directions to the second variation, the authors prove an upper semicontinuity result for the extended Morse index plus nullity under bubble-tree convergence to a limit consisting of a base map and bubbles. The approach combines a refined Hodge decomposition, Lorentz-space estimates, Wente-type inequalities, entropy-type conditions, and spectral diagonalization with respect to neck weights, extending the DGR22 framework to the Sacks-Uhlenbeck setting for maps into and yielding robust Morse-theoretic stability for conformally invariant 2D variational problems.

Abstract

In this paper we consider sequences of -harmonic maps, , from a closed Riemann surface into the -dimensional sphere with uniform bounded energy. These are critical points of the energy Our two main results are an improved pointwise estimate of the gradient in the neck regions around blow up points and the proof that the necks are asymptotically not contributing to the negativity of the second variation of the energy This allows us, in the spirit of the paper of the first and second authors in collaboration with M. Gianocca {\em Morse index stability for critical points to conformally invariant Lagrangians}, to show the upper semicontinuity of the Morse index plus nullity for sequences of -harmonic maps into a sphere.

Paper Structure

This paper contains 16 sections, 39 theorems, 373 equations.

Table of Contents

  1. Introduction
  2. Preliminary definition and results
  3. Energy quantization
  4. Analysis of $(1+\left\vert\nabla u_k\right\vert^2)^{\frac{p_k}{2}-1} (u_k\wedge\nabla u_k)$
  5. $L^{2}$-energy quantization
  6. $L^{2,1}$-energy quantization
  7. Pointwise Control of the Gradient in the Neck Regions
  8. Stability of the Morse Index
  9. Positive contribution of the Necks
  10. The Diagonalization of $Q_{u_k}$ with respect to the Weights $\omega_{\eta,k}$
  11. Appendix
  12. Whitney Type Extension
  13. Monotonicity Formula
  14. Hodge/Helmholtz-Weyl Decomposition
  15. Weighted Poincarè Inequality
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $u_{k}:(\Sigma,h)\to\mathbb{S}^n$ be a sequence of critical points of $E_{p_k}$ with uniformly bounded energy $E_{p_k}(u_k)\leq \Lambda$ and assume that it converges in the bubble tree sense to $(u_\infty, v_\infty^1\ldots v_\infty^Q)$. Then for $k$ large enough,

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: $p-$Harmonic Map
  • Lemma 2.2: $\varepsilon$-regularity
  • Proposition 2.3: Second variation of $p$-energy
  • proof
  • Definition 2.4: Morse index of $p$-harmonic maps
  • Definition 2.5: Bubble tree convergence with one bubble
  • Lemma 3.1
  • proof
  • ...and 65 more