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Non-strict plurisubharmonicity of energy on Teichmüller space

Ognjen Tošić

Abstract

For an irreducible representation $ρ:π_1(Σ_g)\to\mathrm{GL}(n,\mathbb{C})$ there is an energy functional $\mathrm{E}_ρ:\mathcal{T}_g\to\mathbb{R}$, defined on Teichmüller space by taking the energy of the associated equivariant harmonic map into the symmetric space $\mathrm{GL}(n,\mathbb{C})/\mathrm{U}(n)$. It follows from a result of Toledo that $\mathrm{E}_ρ$ is plurisubharmonic, i.e. its Levi form is positive semi-definite. We study the kernel of this Levi form, and relate it to the $\mathbb{C}^*$ action on the moduli space of Higgs bundles. We also show that the points in $\mathcal{T}_g$ where strict plurisubharmonicity fails (i.e. this kernel is non-zero) are critical points of the Hitchin fibration. When $n\geq 2$ and $g\geq 3$, we show that for a generic choice $(S,ρ)$, the map $\mathrm{E}_ρ$ is strictly plurisubharmonic. We also describe the kernel of the Levi form for $n=1$.

Non-strict plurisubharmonicity of energy on Teichmüller space

Abstract

For an irreducible representation there is an energy functional , defined on Teichmüller space by taking the energy of the associated equivariant harmonic map into the symmetric space . It follows from a result of Toledo that is plurisubharmonic, i.e. its Levi form is positive semi-definite. We study the kernel of this Levi form, and relate it to the action on the moduli space of Higgs bundles. We also show that the points in where strict plurisubharmonicity fails (i.e. this kernel is non-zero) are critical points of the Hitchin fibration. When and , we show that for a generic choice , the map is strictly plurisubharmonic. We also describe the kernel of the Levi form for .
Paper Structure (41 sections, 21 theorems, 135 equations)

This paper contains 41 sections, 21 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Results on Riemannian manifolds with very strongly seminegative curvature
  3. Outline and organization
  4. Riemannian manifolds
  5. Higgs bundles
  6. Preliminaries
  7. Non-abelian Hodge correspondence
  8. The Hitchin fibration
  9. Hodge theory on holomorphic vector bundles
  10. Symmetric spaces
  11. Maurer--Cartan form
  12. Levi form of the energy
  13. First-order variation of the harmonic map equation
  14. Proof of Theorem \ref{['thm:lap-energy-zero']} assuming Theorem \ref{['thm:lap-energy']}
  15. Proof of Theorem \ref{['thm:lap-energy']}
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $\rho:\pi_1(\Sigma_g)\to\mathrm{GL}(n,\mathbb{C})$ be an irreducible representation. Then the space $K_\rho$ of directions in $T\mathcal{T}_g$ in which $\mathrm{E}_\rho$ is not strictly plurisubharmonic is exactly the space of directions annihilated by the derivative of $\mathcal{R}_\rho$.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Proposition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.7
  • Definition 1.8
  • Theorem 1.9
  • ...and 43 more