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Yamabe problems for formally self-adjoint, conformally covariant, polydifferential operators

Jeffrey S. Case

Abstract

Formally self-adjoint, conformally covariant, polydifferential operators provide a general framework for studying variational problems, such as prescribing the scalar, $Q$-, or $σ_2$-curvatures, within a conformal class. We describe recent progress on Yamabe problems for such operators, including uniqueness results on the sphere and nonuniqueness results in general. We also highlight a number of open questions related to these operators, some of which constitute a possible blueprint for the general solution of the Yamabe problem for polydifferential operators.

Yamabe problems for formally self-adjoint, conformally covariant, polydifferential operators

Abstract

Formally self-adjoint, conformally covariant, polydifferential operators provide a general framework for studying variational problems, such as prescribing the scalar, -, or -curvatures, within a conformal class. We describe recent progress on Yamabe problems for such operators, including uniqueness results on the sphere and nonuniqueness results in general. We also highlight a number of open questions related to these operators, some of which constitute a possible blueprint for the general solution of the Yamabe problem for polydifferential operators.
Paper Structure (7 sections, 15 theorems, 124 equations, 1 table)

This paper contains 7 sections, 15 theorems, 124 equations, 1 table.

Table of Contents

  1. Introduction
  2. Conformally covariant polydifferential operators
  3. Weyl operators
  4. The Frank--Lieb argument
  5. Geometric Aubin sets
  6. A stronger nonuniqueness result
  7. Future directions

Key Result

Theorem 1.1

Let $I$ be a CVI of homogeneity $-2k \not= -n$ on $n$-manifolds. There is a minimal integer $r \leq 2k$ for which there exists a formally self-adjoint, conformally covariant, polydifferential operator $D$ of rank $r$ satisfying Condition eqn:associated-operator. Moreover, $D$ is unique among all suc

Theorems & Definitions (54)

  • Theorem 1.1
  • Lemma 1.2
  • Definition 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 44 more