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Conformally compact and higher conformal Yang-Mills equations

A. Rod Gover, Emanuele Latini, Andrew Waldron, Yongbing Zhang

Abstract

On conformally compact manifolds we study Yang-Mills equations, their boundary conditions, formal asymptotics, and Dirichlet-to-Neumann maps. We find that smooth solutions with "magnetic" Dirichlet boundary data are obstructed by a conformally invariant, higher order boundary current. We study the asymptotics of the interior Yang-Mills energy functional and show that the obstructing current is the variation of the conformally invariant coefficient of the first log term in this expansion which is a higher derivative conformally invariant analog of the Yang-Mills energy. The invariant energy is the anomaly for the renormalized interior Yang-Mills functional and its variation gives higher conformal Yang-Mills equations. Global solutions to the magnetic boundary problem determine higher order "electric" Neumann data. This yields the Dirichlet-to-Neumann map. We also construct conformally invariant, higher transverse derivative boundary operators. Acting on interior connections, they give obstructions to solving the Yang-Mills boundary problem, determine the asymptotics of formal solutions, and yield conformally invariant tensors capturing the (non-local) electric Neumann data. We also characterize a renormalized Yang-Mills action functional that encodes global features analogously to the renormalized volume for Poincaré-Einstein structures.

Conformally compact and higher conformal Yang-Mills equations

Abstract

On conformally compact manifolds we study Yang-Mills equations, their boundary conditions, formal asymptotics, and Dirichlet-to-Neumann maps. We find that smooth solutions with "magnetic" Dirichlet boundary data are obstructed by a conformally invariant, higher order boundary current. We study the asymptotics of the interior Yang-Mills energy functional and show that the obstructing current is the variation of the conformally invariant coefficient of the first log term in this expansion which is a higher derivative conformally invariant analog of the Yang-Mills energy. The invariant energy is the anomaly for the renormalized interior Yang-Mills functional and its variation gives higher conformal Yang-Mills equations. Global solutions to the magnetic boundary problem determine higher order "electric" Neumann data. This yields the Dirichlet-to-Neumann map. We also construct conformally invariant, higher transverse derivative boundary operators. Acting on interior connections, they give obstructions to solving the Yang-Mills boundary problem, determine the asymptotics of formal solutions, and yield conformally invariant tensors capturing the (non-local) electric Neumann data. We also characterize a renormalized Yang-Mills action functional that encodes global features analogously to the renormalized volume for Poincaré-Einstein structures.
Paper Structure (24 sections, 43 theorems, 406 equations)

This paper contains 24 sections, 43 theorems, 406 equations.

Table of Contents

  1. Introduction
  2. Riemannian and Vector Bundle Conventions
  3. Conformal Geometry
  4. Conformally Compact Manifolds
  5. Conformal Variations and Basic Invariants
  6. The Conformally Compact Yang--Mills Equation
  7. Maxwell System on Hyperbolic Space
  8. Boundary Problems and Asymptotics
  9. Magnetic Problem
  10. Log Solutions
  11. Formal Electric Problem
  12. Poincaré--Einstein Manifolds
  13. Canonical Expansions
  14. Scalar Poincaré--Einstein Boundary Operators
  15. Poincaré--Einstein Asymptotically Yang--Mills Connections
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $d\geq 3$ and $(M,\boldsymbol{c},\sigma)$ be a conformally compact structure. Then when $d\geq 4$ there exists a solution of Problem yeswehavethem to order $\ell=d-4$. When $d=3$ there exists an order $\ell=\infty$ solution.

Theorems & Definitions (99)

  • Theorem 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Theorem 1.11
  • Definition 2.1
  • ...and 89 more