[Paper Review] On the Regularity of Geodesics in the Space of Kähler Metrics

Jianchun Chu, Valentino Tosatti+1 more|arXiv (Cornell University)|Dec 1, 2017

Geometry and complex manifolds30 references12 citations

TL;DR

This paper establishes the existence of $C^{1,1}$ geodesic segments connecting any two Kähler potentials on a compact Kähler manifold, relying on a new a priori interior real Hessian bound for solutions of the nondegenerate complex Monge-Ampère equation that remains valid without requiring a positive lower bound on the right-hand side.

ABSTRACT

We prove that any two Kahler potentials on a compact Kahler manifold can be connected by a geodesic segment of $$C^{1,1}$$ regularity. This follows from an a priori interior real Hessian bound for solutions of the nondegenerate complex Monge-Ampere equation, which is independent of a positive lower bound for the right hand side.

Motivation & Objective

• To establish the regularity of geodesics in the space of Kähler metrics on compact Kähler manifolds.
• To address the challenge of controlling curvature growth in geodesic segments without assuming a positive lower bound on the right-hand side of the complex Monge-Ampère equation.
• To derive a priori interior real Hessian bounds independent of such lower bounds, enabling regularity results under minimal assumptions.
• To extend the understanding of geodesic paths in infinite-dimensional Kähler geometry by proving $C^{1,1}$ regularity for all connecting paths between Kähler potentials.

Proposed method

• Derives an a priori interior real Hessian bound for solutions of the nondegenerate complex Monge-Ampère equation.
• Applies the bound in the context of geodesic equations in the space of Kähler metrics, focusing on $C^{1,1}$ regularity.
• Uses techniques from elliptic PDE theory and complex geometry to control second-order derivatives of Kähler potentials along geodesics.
• Establishes the bound uniformly across the interior of the domain, independent of the infimum of the right-hand side of the equation.
• Relies on maximum principle arguments and careful estimates on the Hessian of the solution to the complex Monge-Ampère equation.
• Constructs a geodesic segment between any two Kähler potentials by solving the geodesic equation with the derived regularity control.

Experimental results

Research questions

• RQ1 Can geodesic segments connecting any two Kähler potentials on a compact Kähler manifold be shown to possess $C^{1,1}$ regularity?
• RQ2 Is it possible to derive a priori real Hessian bounds for solutions of the nondegenerate complex Monge-Ampère equation without assuming a positive lower bound on the right-hand side?
• RQ3 Does the absence of such a lower bound constraint affect the regularity of geodesics in the space of Kähler metrics?
• RQ4 Can the regularity of geodesics be established under minimal assumptions on the data, using intrinsic geometric and analytic tools?
• RQ5 What is the role of interior Hessian estimates in controlling the behavior of geodesics in infinite-dimensional Kähler manifolds?

Key findings

• Any two Kähler potentials on a compact Kähler manifold can be connected by a geodesic segment of $C^{1,1}$ regularity.
• An a priori interior real Hessian bound is established for solutions of the nondegenerate complex Monge-Ampère equation.
• The Hessian bound is independent of a positive lower bound on the right-hand side of the equation, which is a key novelty.
• The bound ensures uniform control over second-order derivatives of the Kähler potential along geodesics.
• The result implies that geodesics in the space of Kähler metrics are well-behaved in the $C^{1,1}$ topology regardless of the size of the right-hand side.
• The method provides a robust framework for analyzing geodesic paths in Kähler geometry without restrictive assumptions on curvature or volume forms.