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Conformal Fractional Dirac Operator and Spinorial Q-curvature

Ali Maalaoui

TL;DR

The paper develops the theory of the conformal fractional Dirac operator $$ on spin manifolds and its associated fractional spinorial Yamabe problem, leveraging a Caffarelli-Silvestre type extension to view the operator as a Dirichlet-to-Neumann map. It furnishes energy identities and a weighted Sobolev inequality for spinors, linking bulk and boundary energies through a weighted extension on Poincaré–Einstein manifolds. In the critical case $ u=n/2$, it introduces a spinorial Q-curvature operator $$ acting on the kernel of the critical operator and studies its conformal variation, yielding a conformal invariant scalar quantity. The work extends scalar fractional conformal geometry to spin geometry, providing explicit expressions on the sphere and Euclidean space and establishing a framework for future analysis of spinorial nonlocal equations and invariants in conformal geometry.

Abstract

In this paper we introduce the conformal fractional Dirac operator and its associated fractional spinorial Yamabe problem. We also present a Caffarelli-Silvestre type extension for this fractional operator, allowing us to express it as a Dirichlet-to-Neumann type operator. As a consequence, we exhibit energy inequalities associated to this operator along with a weighted type Sobolev inequality for spinors. In the second part of the paper, we focus on the critical operator (which can be local or non-local depending on the evenness of the dimension). We introduce a Q-curvature operator, acting on spinors generalizing the classical notion of the scalar Q-curvature.

Conformal Fractional Dirac Operator and Spinorial Q-curvature

TL;DR

The paper develops the theory of the conformal fractional Dirac operator on spin manifolds and its associated fractional spinorial Yamabe problem, leveraging a Caffarelli-Silvestre type extension to view the operator as a Dirichlet-to-Neumann map. It furnishes energy identities and a weighted Sobolev inequality for spinors, linking bulk and boundary energies through a weighted extension on Poincaré–Einstein manifolds. In the critical case , it introduces a spinorial Q-curvature operator acting on the kernel of the critical operator and studies its conformal variation, yielding a conformal invariant scalar quantity. The work extends scalar fractional conformal geometry to spin geometry, providing explicit expressions on the sphere and Euclidean space and establishing a framework for future analysis of spinorial nonlocal equations and invariants in conformal geometry.

Abstract

In this paper we introduce the conformal fractional Dirac operator and its associated fractional spinorial Yamabe problem. We also present a Caffarelli-Silvestre type extension for this fractional operator, allowing us to express it as a Dirichlet-to-Neumann type operator. As a consequence, we exhibit energy inequalities associated to this operator along with a weighted type Sobolev inequality for spinors. In the second part of the paper, we focus on the critical operator (which can be local or non-local depending on the evenness of the dimension). We introduce a Q-curvature operator, acting on spinors generalizing the classical notion of the scalar Q-curvature.
Paper Structure (14 sections, 28 theorems, 193 equations)

This paper contains 14 sections, 28 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Motivation and Preliminaries
  3. The Spinorial Yamabe Problem
  4. The scattering operator and fractional Laplacian
  5. Spinorial Geometry and the Scattering Operator
  6. The Conformal Fractional Dirac Operator and The Spinorial Yamabe Problem
  7. The Extension Problem
  8. Energy Identities
  9. General Energy Inequalities
  10. Sobolev-type Inequality
  11. Critical Operator and $Q$-curvature
  12. Appendix
  13. The case of the flat space $\mathbb{R}^{n}$
  14. The case of the round sphere

Key Result

Theorem 2.1

Let $f\in C^{\infty}(M)$ and $\gamma \in (0,\frac{n}{2})$ then the eigenvalue problem has a unique solution of the form where $F, G, A, B \in C^{\infty}(\overline{X})$ and $F|_{M}=A|_{M}=f$.

Theorems & Definitions (36)

  • Theorem 2.1: GZ
  • Proposition 2.1: GZ
  • Definition 2.1
  • Proposition 2.2: GMPGMP2
  • Theorem 2.2: GMPGMP2
  • Proposition 2.3: GMP
  • Theorem 2.3: GMP2F
  • Proposition 2.4
  • Definition 2.2
  • Proposition 2.5
  • ...and 26 more