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Singular Solutions for the Conformal Dirac-Einstein Problem on the Sphere

Ali Maalaoui, Vittorio Martino, Tian Xu

Abstract

In this paper we investigate the existence of singular solutions to the conformal Dirac-Einstein system. Because of its conformal invariance, there are many similarities with the classical construction of singular solutions for the Yamabe problem. We construct here a family of singular solutions, on the three-dimensional sphere, having exactly two singularities.

Singular Solutions for the Conformal Dirac-Einstein Problem on the Sphere

Abstract

In this paper we investigate the existence of singular solutions to the conformal Dirac-Einstein system. Because of its conformal invariance, there are many similarities with the classical construction of singular solutions for the Yamabe problem. We construct here a family of singular solutions, on the three-dimensional sphere, having exactly two singularities.
Paper Structure (5 sections, 13 theorems, 132 equations)

This paper contains 5 sections, 13 theorems, 132 equations.

Table of Contents

  1. Introduction and main results
  2. Geometric and Analytical Settings
  3. Proof of the Main Theorem
  4. Small Oscillation and Periodic Orbits
  5. Solutions with Large Period

Key Result

Theorem 1.1

Let $T_0=2^{\frac{3}{4}}\pi$. Then there exist $T_{0}<T_{1}\leq T_{2}$ such that for $T\in (T_0,T_{1})\cup (T_{2},+\infty)$, there exists a family $(u_{T},v_{T},a_{T},b_{T})$ of non-constant $2T$-periodic solutions to $(ham)$. Moreover,

Theorems & Definitions (24)

  • Theorem 1.1
  • Corollary 1.1
  • Lemma 3.1
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.2
  • proof
  • ...and 14 more