[Paper Review] On the space of Kahler potentials

Weiyong He|arXiv (Cornell University)|Aug 5, 2012

Geometry and complex manifolds7 references6 citations

TL;DR

This paper establishes the existence and uniqueness of a geodesic segment in the space of generalized Kahler potentials with uniformly bounded mixed second derivatives, extending Chen's foundational result to the broader class of $\mathcal{H}_{1,1}$ potentials. By proving a priori estimates for a regularized complex Monge-Ampere equation that are independent of the lower bound on $n + \Delta\phi$, the author constructs a geodesic whose mixed derivatives remain uniformly controlled, resolving a key regularity question in Kahler geometry.

ABSTRACT

We consider the geodesic equation for the generalized Kahler potential with only mixed second derivatives bounded. We show that given such two generalized Kahler potentials, there is a unique geodesic segment such that for each point on the geodesic, the generalized Kahler potential has uniformly bounded mixed second derivatives (in manifold directions). This generalizes a fundamental theorem of Chen \cite{Chen00} on the space of Kahler potentials.

Motivation & Objective

• To extend Chen's theorem on geodesics in the space of smooth Kahler potentials to the larger space of generalized $C^{1,1}$ potentials ($\mathcal{H}_{1,1}$).
• To establish uniform a priori estimates for the complex Monge-Ampere equation that are independent of the lower bound on $n + \Delta\phi$ at the endpoints.
• To prove the existence of a unique geodesic segment in $\mathcal{H}_{1,1}$ such that $n + \Delta\phi(t)$ remains uniformly bounded for all $t \in [0,1]$.
• To address the open question of whether $\mathcal{H}_{1,1}$ (or $\mathcal{H}_{\infty}$) is a metric space with nonpositive curvature, by providing a key regularity result.

Proposed method

• Regularize the geodesic equation by introducing a right-hand side $\epsilon e^f$ to ensure smooth solutions for $\epsilon > 0$, enabling the use of standard elliptic estimates.
• Apply a modified version of Yau's $C^2$ estimate technique to the regularized equation, focusing on controlling $\Delta\phi$ directly rather than $\phi_{tt} + \Delta\phi$.
• Leverage the product structure $M \times [0,1]$ to separate manifold and time directions, allowing independent control of mixed derivatives.
• Use a maximum principle argument on the quantity $\log(n + \Delta\phi) - C\phi + t^2$ to derive the key $L^\infty$ bound on $\Delta\phi$.
• Pass to the limit as $\epsilon \to 0$ using uniform estimates independent of the lower bound on $n + \Delta\phi_0$ and $n + \Delta\phi_1$, yielding a generalized solution.
• Construct the geodesic as the $C^\alpha$ limit of smooth solutions $\phi^k$ approximating the endpoints $\phi_0, \phi_1 \in \mathcal{H}_{1,1}$.

Experimental results

Research questions

• RQ1 Does a unique geodesic exist in the space of generalized $C^{1,1}$ Kahler potentials $\mathcal{H}_{1,1}$ with uniformly bounded mixed second derivatives?
• RQ2 Can uniform a priori estimates for $\Delta\phi$ be obtained in the geodesic equation without requiring a positive lower bound on $n + \Delta\phi$ at the endpoints?
• RQ3 Is the space $\mathcal{H}_{1,1}$ a metric space under Mabuchi's metric, and does it inherit nonpositive curvature from $\mathcal{H}$?
• RQ4 What is the relationship between $\mathcal{H}_{1,1}$ and the metric completion of the smooth space $\mathcal{H}$?
• RQ5 Can the regularity of geodesics in $\mathcal{H}_{1,1}$ be used to answer deeper questions about the structure of $\mathcal{H}_{\infty}$ and its metric properties?

Key findings

• For any $\phi_0, \phi_1 \in \mathcal{H}_{1,1}$, there exists a unique generalized solution $\phi(t)$ to the geodesic equation (1.1) such that $n + \Delta\phi(t)$ is uniformly bounded for all $t \in [0,1]$.
• The uniform bound on $n + \Delta\phi(t)$ depends only on $n$, $\|\phi_i\|_{L^\infty}$, $\sup \Delta\phi_i$, and the geometry of $(M, \omega)$, not on the lower bound of $n + \Delta\phi_i$.
• The a priori estimate for $\Delta\phi$ in Theorem 1.3 is independent of the strictly positive lower bound of $n + \Delta\phi_0$ and $n + \Delta\phi_1$, which was the key missing ingredient in Chen's original proof.
• The proof establishes that $\Delta\phi$ remains uniformly bounded along the geodesic by using a maximum principle on a modified quantity involving $\log(n + \Delta\phi)$ and a quadratic term in $t$, avoiding reliance on $\phi_{tt}$ estimates.
• The geodesic is constructed as the $C^\alpha$ limit of smooth solutions $\phi^k$ to the regularized equation with right-hand side $\frac{1}{k}e^f$, ensuring convergence and uniqueness.
• The result confirms a problem posed by X.-X. Chen and provides a crucial step toward understanding whether $\mathcal{H}_{1,1}$ is a metric space with nonpositive curvature.