Kähler-Ricci Flow preserves negative anti-bisectional curvature

Technical Summary: Kähler-Ricci Flow Preserves Negative Anti-Bisectional Curvature

Problem Statement
This paper investigates the behavior of the "anti-bisectional curvature" under the Kähler-Ricci flow. The anti-bisectional curvature, defined as $R(V, W, V, W)$ for polarized $(1,0)$-vectors $V$ and $W$, arises from a correspondence between the regularity theory of optimal transport and the geometry of Kähler manifolds on tube domains. Specifically, the Ma-Trudinger-Wang (MTW) tensor for cost functions of the form $c(x, y) = \Psi(x-y)$ corresponds to the anti-bisectional curvature of an associated Kähler metric.

While the Ricci flow is known to preserve various positivity conditions on curvature (e.g., non-negative bisectional curvature), it is generally not known to preserve negativity conditions, with the notable exception of negative Gaussian curvature for Riemann surfaces. The authors address the question of whether non-positive anti-bisectional curvature is preserved under the flow, a property that would have significant implications for the regularity of optimal transport maps and the geometry of Kähler-Einstein metrics.

Methodology
The authors study the Kähler-Ricci flow on tube domains $T\Omega = \Omega + i\mathbb{R}^n$, where $\Omega \subset \mathbb{R}^n$ is a convex domain. These domains correspond to the tangent bundles of Hessian manifolds equipped with Kähler-Sasaki metrics. The flow is analyzed via the evolution of the curvature tensor.

1. Uhlenbeck's Trick: To simplify the complex reaction-diffusion equations governing the curvature evolution, the authors employ Uhlenbeck's trick. They introduce an abstract vector bundle isomorphic to the complex tangent bundle and a time-dependent isometry to transform the curvature tensor into a form where the evolution equation simplifies to a reaction-diffusion system with a quadratic reaction term $Q(\tilde{R})$.
2. Maximum Principle: Since the manifolds considered are often non-compact (tube domains), the standard maximum principle for compact manifolds is insufficient. The authors utilize a version of the maximum principle for complete manifolds with bounded curvature, originally established by Shi. This requires verifying a "null-vector condition" for the reaction term $Q$.
3. Null-Vector Verification: The core technical contribution involves proving that if the anti-bisectional curvature is non-positive (or non-negative for the orthogonal case in dimension two) and vanishes on a specific pair of vectors, the reaction term $Q$ is "inward-pointing" (i.e., non-positive or non-negative, respectively) at that point. This verification relies on algebraic manipulations of the curvature tensor symmetries and specific lemmas regarding semi-positive definite matrices (adapted from work by Cao and Hamilton).
4. Dimensional Constraints: The analysis distinguishes between general dimensions and complex dimension two. For the preservation of non-negative orthogonal anti-bisectional curvature, the proof relies on specific algebraic cancellations that only hold when the complex dimension $n=2$.

Key Contributions and Results

• Theorem 1 (Preservation of Non-Positive Anti-Bisectional Curvature): The authors prove that if a complete Kähler manifold with bounded curvature has non-positive anti-bisectional curvature, this condition is preserved under the Kähler-Ricci flow for all time the flow exists. This is a significant result as it identifies a rare instance of a preserved negativity condition in higher-dimensional Ricci flow.
• Theorem 2 (Preservation of Non-Negative Orthogonal Anti-Bisectional Curvature in Dimension 2): In complex dimension two, the authors show that non-negative orthogonal anti-bisectional curvature (the MTW(0) condition restricted to orthogonal vectors) is also preserved under the flow. They note that this result does not generalize to dimensions $n \geq 3$ due to the sign of the quadratic reaction term.
• Geometric Consequences (Corollaries 1–4):
  • Kähler-Einstein Metrics: Combining Theorem 1 with Tong's convergence results for metrics with negatively pinched holomorphic sectional curvature, the authors show that the resulting Kähler-Einstein metric on a tube domain retains non-positive anti-bisectional curvature.
  • Negative Cost-Curvature: For smooth, bounded, and strongly convex domains, the Kähler-Einstein metric is shown to have "negative cost-curvature" (a stronger condition implying non-positive anti-bisectional curvature and negatively pinched polarized holomorphic sectional curvature).
  • Curvature Control: The authors derive quantitative estimates showing that for positive time $t$, the polarized holomorphic sectional curvature is bounded below by a term depending on the orthogonal anti-bisectional trace curvature ($O$) and the initial scalar curvature bound, independent of the initial holomorphic sectional curvature pinching.
• Applications to Optimal Transport:
  • KR Weak Regularity: The preservation of non-negative orthogonal anti-bisectional curvature in dimension 2 allows for the definition of "KR weakly regular" cost functions. These are cost functions $c(x,y) = \Psi(x-y)$ where the potential $\Psi$ evolves under a parabolic flow to maintain the MTW(0) condition.
  • Regularity Estimates: Using this flow, the authors establish interior Hölder estimates for optimal transport maps associated with $W^{2,p}$ convex potentials in two dimensions, generalizing existing regularity theory to less smooth cost functions.

Significance and Claims
The paper claims that the preservation of non-positive anti-bisectional curvature is "somewhat unexpected," as most known invariant curvature conditions under Ricci flow are positivity conditions. The authors emphasize that this result provides a new tool for studying the relationship between holomorphic sectional curvature and Ricci curvature in complex geometry.

In the context of optimal transport, the work bridges the gap between the geometric evolution of metrics and the regularity of transport maps. By showing that the flow preserves the MTW(0) condition (in dimension 2), the paper offers a canonical method to deform cost functions to ensure regularity, potentially extending regularity theory to rougher cost functions (such as Monge's original cost $|x-y|$) via approximation.

The authors also provide concrete examples of metrics satisfying these conditions, including Calabi's metric on the tube domain over the unit ball and metrics on cones, demonstrating that the assumption of negative anti-bisectional curvature is not vacuous. Finally, the paper notes that while the results are robust for complete metrics, the extension to incomplete metrics (common in optimal transport) remains an open question requiring suitable boundary conditions.