A Metric Lower Bound Estimate for Geodesics in the Space of Kähler Potentials

1. Problem Statement and Context

Background:
The paper addresses the geometry of the space of Kähler potentials, denoted as $\mathcal{H} = \{\phi \in C^\infty(V) \mid \omega_0 + \sqrt{-1}\partial\bar{\partial}\phi > 0\}$, on a compact Kähler manifold $(V, \omega_0)$. This space is equipped with a Riemannian metric (Mabuchi metric). A central problem in Kähler geometry is the existence and regularity of geodesics connecting two points $\phi_0, \phi_1 \in \mathcal{H}$.

The Geodesic Equation:
Geodesics in this infinite-dimensional space correspond to solutions of the homogeneous complex Monge-Ampère (HCMA) equation on a product space $S \times V$ (where $S$ is a strip or a Riemann surface):
$$(\Omega_0 + \sqrt{-1}\partial\bar{\partial}\Phi)^{n+1} = 0$$
subject to Dirichlet boundary conditions.

The Core Difficulty:
While the global $C^{1,1}$ regularity of weak solutions is well-established, a major open issue is whether the solution $\Phi$ maintains strict positivity of the metric along the geodesic. Specifically, does $\omega_0 + \sqrt{-1}\partial\bar{\partial}\Phi(\tau, \cdot) > 0$ hold for all $\tau$ in the domain?

• Previous counterexamples (e.g., by Stoppa, Chen, and others) show that for general boundary data, the metric can degenerate (rank drop) or even vanish on open sets.
• The paper aims to prove that if the boundary data is sufficiently close (in $C^2$ norm) to the background metric, the metric along the geodesic remains strictly positive and bounded away from zero, independent of the regularization parameter.

2. Methodology

The author employs a continuity method combined with maximum principle estimates on a family of non-degenerate approximations.

A. Approximation Scheme

Instead of solving the degenerate HCMA equation directly, the author considers the non-degenerate Dirichlet problem (Problem 1.3) on a domain $R \times V$:
$$(\Omega_0 + \sqrt{-1}\partial\bar{\partial}\Phi) ^{n+1} = \epsilon \sqrt{-1} d\tau \wedge d\bar{\tau} \wedge \Omega_0^n$$
where $\epsilon > 0$ is a small parameter.

• For $\epsilon > 0$, a smooth solution exists with $\omega_0 + \sqrt{-1}\partial\bar{\partial}\Phi > 0$.
• The goal is to establish a lower bound for the metric that does not vanish as $\epsilon \to 0$.

B. The Key Quantity $Q$

To control the second-order derivatives of $\Phi$ (which determine the metric), the author constructs a specific auxiliary quantity $Q$:
$$Q = \Phi_{,\alpha\beta}\Phi_{,\gamma\theta}g^{\alpha\theta}g^{\beta\gamma} + \Phi_{\alpha\beta}\Phi_{\gamma\theta}g^{\alpha\theta}g^{\gamma\beta} + \Phi_{\alpha}\Phi_{\beta}g^{\alpha\beta}$$
Here, $\Phi_{,\alpha\beta}$ denotes covariant derivatives with respect to the background metric $\omega_0$, and $\Phi_{\alpha\beta}$ denotes mixed derivatives involving the $\tau$ variable. The quantity $Q$ essentially measures the deviation of the solution's Hessian from the background metric.

C. The Operator $L$

The author defines a linear elliptic operator $L$ (a scalar multiple of the Laplacian associated with the solution metric) and computes $LQ$.

• Computation: The paper involves extensive tensor calculus to derive equations for $L(\Phi_{,\alpha\beta})$, $L(\Phi_{\alpha\beta})$, and $L(\Phi_\theta)$.
• Curvature Terms: The computation accounts for curvature terms ($R$) and their covariant derivatives arising from the background manifold $V$.
• Inequality Derivation: The author shows that under the assumption that $Q$ is small (specifically $Q < 1$), the operator satisfies:
  $$LQ \geq -\lambda Q$$
  for a small constant $\lambda$ depending on $\epsilon$ and the geometry of $V$.

D. Continuity Argument

Using the inequality $LQ \geq -\lambda Q$, the author applies the maximum principle to derive an a priori estimate:
$$\max_{R \times V} Q \leq C \max_{\partial R \times V} Q$$
By assuming the boundary data $F$ has a small $C^2$ norm (implying $Q$ is small on the boundary), the author uses a continuity argument (varying the boundary data from 0 to $F$) to prove that $Q$ remains small throughout the interior. This ensures the metric remains non-degenerate.

3. Key Contributions and Results

Main Theorem 1: Metric Lower Bound for Approximate Solutions

Theorem 1.1: Given a Kähler manifold $(V, \omega_0)$ and a bounded domain $R \subset \mathbb{C}$, there exists a constant $\hat{\delta}$ (depending on the dimension and curvature of $V$) such that if:

1. The parameter $\epsilon$ is sufficiently small: $0 < \epsilon < \frac{\hat{\delta}}{(\text{diam } R)^2}$.
2. The boundary data $F$ has a small $C^2$ norm: $|F|_{C^2} < \hat{\delta}$.

Then the solution $\Phi$ to the non-degenerate problem satisfies:
$$\omega_0 + \sqrt{-1}\partial\bar{\partial}\Phi(\tau, \cdot) > \frac{1}{2}\omega_0 \quad \text{for all } \tau \in R.$$
Crucially, this lower bound is independent of $\epsilon$.

Main Theorem 2: Metric Lower Bound for Geodesics

Theorem 1.2: As a consequence of Theorem 1.1, for any two Kähler potentials $\phi_0, \phi_1 \in \mathcal{H}$ with sufficiently small $C^2$ norms (close to each other in the $C^2$ topology), the weak geodesic $\Phi$ connecting them satisfies:
$$\omega_0 + \sqrt{-1}\partial\bar{\partial}\Phi(\tau, \cdot) \geq \frac{1}{2}\omega_0 \quad \text{for all } \tau \in S.$$
This implies that the associated metrics do not degenerate along the geodesic.

Appendix A: 1-Dimensional Flat Torus Case

The author provides a more complete analysis for the case where $V$ is a 1-dimensional flat torus.

• In this case, curvature terms vanish, simplifying the equations significantly.
• The author defines a "cone" structure in the jet space and proves that if the boundary data lies within a specific convex cone (related to $\omega_0$-convexity), the solution remains within that cone.
• This offers a geometric intuition for the general result: the solution stays within a "safe" region of the metric space.

4. Significance

1. Resolution of Degeneracy for Small Perturbations: The paper resolves the question of metric degeneracy for "small" geodesics. While general geodesics may degenerate, this work proves that if the endpoints are close enough in the $C^2$ topology, the geodesic is "well-behaved" (metrics stay uniformly positive).
2. Optimal Regularity vs. Non-Degeneracy: While $C^{1,1}$ is known to be the optimal regularity for general solutions, this paper establishes that under smallness conditions, the solution retains a strong non-degeneracy property, which is stronger than just regularity.
3. Technical Innovation: The construction of the quantity $Q$ and the precise estimation of the operator $LQ$ involving curvature terms provide a robust framework for analyzing the degenerate Monge-Ampère equation. The method avoids relying on specific symmetries of the manifold, making it applicable to general Kähler manifolds.
4. Implications for Kähler Geometry: This result is relevant to the study of the existence of constant scalar curvature Kähler (cscK) metrics and the stability of Kähler manifolds, where the behavior of geodesics in the space of potentials is fundamental. It suggests that the "singular" behavior of geodesics is a phenomenon that only occurs for large separations in the space of potentials.

In summary, Jingchen Hu establishes that the space of Kähler potentials is locally "geodesically convex" in a strong metric sense: sufficiently close points can be connected by a geodesic where the geometry remains strictly positive and non-degenerate.