[Paper Review] Regularity of A Complex Monge-Ampère Equation on Hermitian Manifolds

Xiaolan Nie|arXiv (Cornell University)|Nov 18, 2013

Geometry and complex manifolds16 references3 citations

TL;DR

This paper establishes the smoothness of bounded $ ilde{ ho}$-plurisubharmonic solutions to a complex Monge-Ampère equation on compact Hermitian manifolds, under a condition on the background metric that ensures stability of solutions. By extending parabolic flow estimates and applying the maximum principle, it proves that weak solutions are smooth, generalizing a result of Székelyhidi and Tosatti from Kähler to Hermitian geometry.

ABSTRACT

We obtain higher order estimates for a parabolic flow on a compact Hermitian manifold. As an application, we prove that a bounded $\hatω$-plurisubharmonic solution of an elliptic complex Monge-Ampère equation is smooth under an assumption on the background Hermitian metric $\hatω$. This generalizes a result of Székelyhidi and Tosatti on Kähler manifolds.

Motivation & Objective

• To extend the regularity result of Székelyhidi and Tosatti on Kähler manifolds to compact Hermitian manifolds.
• To establish higher-order estimates for a parabolic flow associated with the complex Monge-Ampère equation on Hermitian manifolds.
• To prove that bounded weak solutions of the elliptic complex Monge-Ampère equation are smooth under a metric condition ensuring solution stability.
• To generalize the smoothing property of the parabolic flow to the non-Kähler setting, accounting for torsion terms in Hermitian geometry.
• To provide a foundation for constructing weak solutions to the Chern-Ricci flow with singular initial Gauduchon metrics on complex surfaces.

Proposed method

• Adapts the parabolic flow approach of Székelyhidi and Tosatti by analyzing the evolution equation $\frac{\partial\varphi}{\partial t} = \log\frac{(\hat{\omega}+\sqrt{-1}\partial\bar{\partial}\varphi)^n}{\hat{\omega}^n} + F(\varphi,z)$ on compact Hermitian manifolds.
• Applies the maximum principle to derive short-time existence and $C^0$ estimates for the parabolic flow, with time bounds depending only on $\sup|\varphi_0|$ and $\sup|\dot{\varphi}_0|$.
• Establishes $C^2$, $C^3$, and $|\operatorname{Ric}|$ estimates by adapting techniques from Gill, Tosatti-Weinkove, and Phong-S\v{\i}sem-Sturm, carefully handling torsion terms in Hermitian geometry.
• Uses parabolic Schauder estimates to obtain $C^{2+\alpha,1+\alpha/2}$ bounds on $\dot{\varphi}$ and $\varphi_k, \varphi_{\bar{k}}$ for $t \in [\epsilon, T']$, then iterates to obtain all higher-order estimates.
• Applies Kołodziej’s stability result under the assumption $\int_M (\hat{\omega}+\sqrt{-1}\partial\bar{\partial}u)^n = \int_M \hat{\omega}^n$ for bounded $\omega$-psh functions to ensure uniqueness and continuity of solutions.
• Constructs a sequence of smooth approximations $\psi_j$ to the weak solution $\phi$, evolves them via the parabolic flow, and shows the limit $\beta(t,z) = \lim \varphi_j$ satisfies $\dot{\beta}(t) = 0$ for $t > 0$, implying $\phi = \beta(0)$ is smooth.

Experimental results

Research questions

• RQ1 Can the regularity result for weak solutions of the complex Monge-Ampère equation on Kähler manifolds be extended to compact Hermitian manifolds?
• RQ2 What conditions on the Hermitian metric $\hat{\omega}$ ensure the stability of solutions necessary for regularity?
• RQ3 How do torsion terms in Hermitian geometry affect the higher-order estimates for the parabolic flow?
• RQ4 Can the parabolic flow approach be used to prove smoothness of weak solutions in the absence of a Kähler structure?
• RQ5 Under what conditions can the Chern-Ricci flow be extended past singular times via weak solutions with singular initial data?

Key findings

• A bounded weak solution $\phi$ of the complex Monge-Ampère equation $(\hat{\omega}+\sqrt{-1}\partial\bar{\partial}\phi)^n = e^{-F(\phi,z)}\hat{\omega}^n$ is smooth on a compact Hermitian manifold if the metric $\hat{\omega}$ satisfies $\int_M (\hat{\omega}+\sqrt{-1}\partial\bar{\partial}u)^n = \int_M \hat{\omega}^n$ for all bounded $\hat{\omega}$-psh functions $u$.
• The parabolic flow $\frac{\partial\varphi}{\partial t} = \log\frac{(\hat{\omega}+\sqrt{-1}\partial\bar{\partial}\varphi)^n}{\hat{\omega}^n} + F(\varphi,z)$ has a smooth solution on $[0,T]$ for some $T>0$ depending only on $\sup|\varphi_0|$ and $\sup|\dot{\varphi}_0|$.
• Higher-order estimates for the parabolic flow are obtained via parabolic Schauder estimates, with bounds blowing up as $t \to 0^+$, but uniform on $[\epsilon,T]$ for any $\epsilon > 0$
• The limit of the parabolic flow starting from smooth approximations $\psi_j$ to $\phi$ satisfies $\dot{\beta}(t) = 0$ for $t > 0$, implying $\beta(t) = \beta(0)$ is smooth, so $\phi = \beta(0)$ is smooth.
• The assumption $\int_M (\hat{\omega}+\sqrt{-1}\partial\bar{\partial}u)^n = \int_M \hat{\omega}^n$ holds, for example, when $\partial\bar{\partial}\hat{\omega}^k = 0$ for $k=1,2$, as in Guan-Li's condition.
• The result suggests that the Chern-Ricci flow may continue past singular times by pushing down the limiting current, supporting a conjecture of Tosatti and Weinkove on non-collapsing flows on complex surfaces.