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A gluing construction of singular solutions for a fully non-linear equation in conformal geometry

María Fernanda Espinal, María del Mar González

Abstract

In this paper we study the $σ_2$--Yamabe equation, $n>4$, for solutions with a prescribed singular set $Λ$ given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than $(n-\sqrt{n}-2)/2$. The $σ_2$--curvature in conformal geometry is defined as the second elementary symmetric polynomial of the eigenvalues of the Schouten tensor, which yields a fully non-linear PDE for the conformal factor. We show that the classical gluing method, used by Mazzeo-Pacard (JDG 1996) for the scalar curvature problem, can be used in the fully non-linear setting. This is a consequence of the conformal properties of the $σ_2$ equation, which imply that the linearized operator has good mapping properties in weighted spaces.

A gluing construction of singular solutions for a fully non-linear equation in conformal geometry

Abstract

In this paper we study the --Yamabe equation, , for solutions with a prescribed singular set given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than . The --curvature in conformal geometry is defined as the second elementary symmetric polynomial of the eigenvalues of the Schouten tensor, which yields a fully non-linear PDE for the conformal factor. We show that the classical gluing method, used by Mazzeo-Pacard (JDG 1996) for the scalar curvature problem, can be used in the fully non-linear setting. This is a consequence of the conformal properties of the equation, which imply that the linearized operator has good mapping properties in weighted spaces.
Paper Structure (29 sections, 26 theorems, 256 equations, 1 figure)

This paper contains 29 sections, 26 theorems, 256 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The model $\mathbb R^n\setminus \mathbb R^p$ in cylindrical coordinates
  3. Cylindrical coordinates
  4. The fast-decay ODE solution
  5. The approximate solution
  6. The model singularity
  7. Fermi coordinates
  8. Construction of $\bar{u}_\epsilon$
  9. The non-linear problem
  10. Explicit calculations in the model case
  11. Indicial roots as $t\to \infty$
  12. Indicial roots as $t\to -\infty$
  13. The linearized operator
  14. The model linearization
  15. A closer look at $\mathbb L_\epsilon$
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $(M,g_M)$ be a $n$-dimensional, compact, smooth, Riemannian manifold of positive $\sigma_2$--curvature (and in the positive $\Gamma_2^+$ cone), that is non-degenerate. Let $\Lambda$ be a subset of $M$ which is a closed, connected, submanifold of dimension $p$. Assume that where $\mathfrak p_2$ is given in exact-formulas below. Then there exists an infinite dimensional family of solutions of e

Figures (1)

  • Figure 1: Our choice of parameters $\mu$ and $\nu$.

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof : Sketch of the proof
  • Corollary 2.2
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Lemma 4.1
  • proof
  • ...and 45 more