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Kähler-Einstein metrics of negative curvature

Henri Guenancia, Ursula Hamenstädt

TL;DR

The paper builds canonical negatively curved Kähler–Einstein metrics on compact nonlocally symmetric complex manifolds by first analyzing invariant KE metrics on Thüllen domains $ Omega_eta$ through an ODE for a transverse growth function $f_eta$, proving exponential convergence to the ball metric away from a divisor, and establishing explicit curvature bounds. It then leverages Stover–Toledo’s arithmetic-congruence construction to obtain covers with arbitrarily large collar neighborhoods around a totally geodesic divisor, enabling a controlled gluing of a model cone KE metric to the ball metric and the subsequent formation of Kähler–Einstein cone metrics on the covers. The glued and cone metrics are shown to maintain uniform negative curvature (and even very strong negativity in certain regimes), producing an infinite family of compact KE manifolds not covered by the ball, with nontrivial holomorphic rigidity. This approach extends canonical negatively curved KE geometry beyond locally symmetric cases and yields a robust toolkit for constructing nonball quotients with controlled curvature properties, including cone metrics on finite covers.

Abstract

Given any integer $n\geq 2$, we construct a compact Kähler-Einstein manifold of dimension n of negative sectional curvature which is not covered by the ball.

Kähler-Einstein metrics of negative curvature

TL;DR

The paper builds canonical negatively curved Kähler–Einstein metrics on compact nonlocally symmetric complex manifolds by first analyzing invariant KE metrics on Thüllen domains through an ODE for a transverse growth function , proving exponential convergence to the ball metric away from a divisor, and establishing explicit curvature bounds. It then leverages Stover–Toledo’s arithmetic-congruence construction to obtain covers with arbitrarily large collar neighborhoods around a totally geodesic divisor, enabling a controlled gluing of a model cone KE metric to the ball metric and the subsequent formation of Kähler–Einstein cone metrics on the covers. The glued and cone metrics are shown to maintain uniform negative curvature (and even very strong negativity in certain regimes), producing an infinite family of compact KE manifolds not covered by the ball, with nontrivial holomorphic rigidity. This approach extends canonical negatively curved KE geometry beyond locally symmetric cases and yields a robust toolkit for constructing nonball quotients with controlled curvature properties, including cone metrics on finite covers.

Abstract

Given any integer , we construct a compact Kähler-Einstein manifold of dimension n of negative sectional curvature which is not covered by the ball.

Paper Structure

This paper contains 14 sections, 20 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Kähler--Einstein metrics on Thüllen domains
  3. Geometric properties of ${\rm U}(n-1,1)$-invariant Kähler metrics on $\Omega_\alpha$
  4. The curvature operator of an invariant Kähler metric
  5. An ordinary differential equation for the Kähler--Einstein metric
  6. The curvature of the Kähler--Einstein metric
  7. Comparison with the pull-back of the ball metric
  8. The construction of Stover and Toledo
  9. Analysis of the Kähler--Einstein cone metric
  10. Ampleness of the adjoint divisor
  11. Kähler--Einstein cone metrics
  12. Cut-off functions
  13. The glued metric
  14. Curvature of the Kähler--Einstein cone metric on $M_R$

Key Result

Theorem 1

For every $n\geq 2$ there exists a compact complex manifold $M$ of dimension $n$ not covered by the ball which admits a Kähler--Einstein metric of negative sectional curvature.

Theorems & Definitions (46)

  • Theorem
  • Lemma 2.1: Naruki
  • proof : Proof of Lemma \ref{['automorphismgroup']}
  • Theorem 2.2: Theorem 7.5 of CY80
  • proof
  • Remark 2.3: Comparison with the Bergman metric
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 36 more