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Conformal symmetries in geometry and harmonic analysis

Bent Ørsted

Abstract

In this essay we give an introduction to conformal symmetry, based on the example of the Yamabe operator and its use in conformal differential geometry, and in representation theory.

Conformal symmetries in geometry and harmonic analysis

Abstract

In this essay we give an introduction to conformal symmetry, based on the example of the Yamabe operator and its use in conformal differential geometry, and in representation theory.
Paper Structure (9 sections, 23 theorems, 111 equations)

This paper contains 9 sections, 23 theorems, 111 equations.

Table of Contents

  1. Overview and motivation
  2. Conformal differential geometry
  3. Asymptotics of heat kernels and their conformal variation
  4. Determinants and their rigidity
  5. Intertwining operators and minimal representations
  6. Constructions of the minimal $O(p,q)$ representation
  7. Branching laws for the minimal $O(p,q)$ representation
  8. The flat model of the minimal representation of $O(p,q)$
  9. Final remarks on branching laws and symmetry--breaking

Key Result

Lemma 1

The Yamabe operator on $M$ of dimension $n$ in the metric $g$ with Laplace operator $\Delta$ and scalar curvature $K$ satisfies with $a = (n-2)/2, \, b = (n+2)/2$. Hence the Yamabe operator $Y$ is conformally covariant with bidegrees $a = (n-2)/2, \, b = (n+2)/2$.

Theorems & Definitions (38)

  • Lemma 1
  • Remark 2
  • Proposition 3
  • Theorem 4
  • Remark 5
  • Theorem 6
  • Theorem 7
  • Remark 8
  • Lemma 9
  • Remark 10
  • ...and 28 more