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Conformally Invariant Dirac Equation with Non-Local Nonlinearity

Ali Maalaoui, Vittorio Martino, Lamine Mbarki

TL;DR

This work analyzes a conformally invariant spinorial equation with a nonlocal convolution nonlinearity, motivated by the conformal Einstein-Dirac problem in dimension $4$. The authors develop a variational framework and establish compactness up to bubbling, energy quantization, and detailed blow-up analysis on manifolds, including a full bubbling decomposition into Euclidean bubbles and ground-state profiles, which leads to an Aubin-type inequality for the spinorial energy. They characterize ground states on the standard sphere, showing that energy minimizers at the critical level correspond to $-1/2$-Killing spinors, and use this to deduce solvability criteria for the conformal Einstein-Dirac problem in dimension $4$ under a strict inequality. Furthermore, they study a linearly perturbed model in the spirit of Brezis-Nirenberg, proving existence of nontrivial ground states for positive perturbations outside the Dirac spectrum, with a Mountain-Pass geometry established via a variational reduction and a carefully constructed test spinor that yields energy below the critical threshold. Overall, the paper extends scalar-conformal techniques to a nonlocal, spinorial setting, providing solvability criteria and new compactness results for conformally invariant nonlocal Dirac-type equations on spin manifolds.

Abstract

We study a conformally invariant equation involving the Dirac operator and a non-linearity of convolution type. This non-linearity is inspired from the conformal Einstein-Dirac problem in dimension 4. We first investigate the compactness, bubbling and energy quantization of the associated energy functional then we characterize the ground state solutions of the problem on the standard sphere. As a consequence, we prove an Aubin-type inequality that assures the existence of solutions to our problem and in particular the conformal Einstein-Dirac problem in dimension 4. Moreover, we investigate the effect of a linear perturbation to our problem, leading us to a Brezis-Nirenberg type result.

Conformally Invariant Dirac Equation with Non-Local Nonlinearity

TL;DR

This work analyzes a conformally invariant spinorial equation with a nonlocal convolution nonlinearity, motivated by the conformal Einstein-Dirac problem in dimension . The authors develop a variational framework and establish compactness up to bubbling, energy quantization, and detailed blow-up analysis on manifolds, including a full bubbling decomposition into Euclidean bubbles and ground-state profiles, which leads to an Aubin-type inequality for the spinorial energy. They characterize ground states on the standard sphere, showing that energy minimizers at the critical level correspond to -Killing spinors, and use this to deduce solvability criteria for the conformal Einstein-Dirac problem in dimension under a strict inequality. Furthermore, they study a linearly perturbed model in the spirit of Brezis-Nirenberg, proving existence of nontrivial ground states for positive perturbations outside the Dirac spectrum, with a Mountain-Pass geometry established via a variational reduction and a carefully constructed test spinor that yields energy below the critical threshold. Overall, the paper extends scalar-conformal techniques to a nonlocal, spinorial setting, providing solvability criteria and new compactness results for conformally invariant nonlocal Dirac-type equations on spin manifolds.

Abstract

We study a conformally invariant equation involving the Dirac operator and a non-linearity of convolution type. This non-linearity is inspired from the conformal Einstein-Dirac problem in dimension 4. We first investigate the compactness, bubbling and energy quantization of the associated energy functional then we characterize the ground state solutions of the problem on the standard sphere. As a consequence, we prove an Aubin-type inequality that assures the existence of solutions to our problem and in particular the conformal Einstein-Dirac problem in dimension 4. Moreover, we investigate the effect of a linear perturbation to our problem, leading us to a Brezis-Nirenberg type result.

Paper Structure

This paper contains 7 sections, 21 theorems, 243 equations.

Table of Contents

  1. Introduction and motivation
  2. Preliminaries
  3. Regularity
  4. Bubbling and energy quantization
  5. Least Energy Solution on the Sphere
  6. Brezis-Nirenberg Problem
  7. Test Spinor

Key Result

Theorem 1.1

Let us assume that $(M,[g])$ has a positive Yamabe constant $Y_s(M,[g])$ and let $(\psi_{k})_{k\in \mathbb{N}}$ be a Palais-Smale sequence for $J_{g}$ at level $c\geq 0$, where $Y_s(M,[g])$ is the s-Yamabe constant, which we define in the next section.. Then there exist $\psi_{\infty}\in C^{\infty}( where with $\sigma_{k,j}=(\rho_{k,j})^{-1}$ and $\rho_{k,j}(\cdot)=exp_{x_{k}^j}(R_{k}^j \cdot)$ i

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Theorem 1.3
  • Proposition 2.1: BourFriedrich2000
  • Proposition 2.2: Chang-GonzalezMarGZ
  • Proposition 2.3
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • ...and 28 more