[Paper Review] A priori $L^{\infty}$-estimates for degenerate complex Monge-Ampère equations

Philippe Eyssidieux, Vincent Guedj+1 more|ArXiv.org|Dec 21, 2007

Geometry and complex manifolds4 references4 citations

TL;DR

This paper establishes uniform $L^{inity}$-estimates for solutions to degenerate complex Monge-Ampère equations on compact Kähler manifolds, where the cohomology class degenerates to a non-big class. Using capacity comparison and Hölder's inequality, the authors prove that solutions remain uniformly bounded in $L^{inity}$-norm as the parameter $t \to 0^+$, generalizing prior results without geometric assumptions on the fibration.

ABSTRACT

We study families of complex Monge-Ampère equations, focusing on the case where the cohomology classes degenerate to a non big class. We establish uniform a priori $L^{\infty}$-estimates for the normalized solutions, generalizing the recent work of S. Kolodziej and G. Tian. This has interesting consequences in the study of the Kähler-Ricci flow.

Motivation & Objective

• To establish uniform a priori $L^\infty$-estimates for solutions to degenerate complex Monge-Ampère equations as the cohomology class degenerates to a non-big class.
• To generalize recent results by Kolodziej and Tian by removing technical geometric assumptions on the fibration $\pi: X \to Y$.
• To provide a uniform bound on the $L^\infty$-norm of normalized solutions $\varphi_t$ independent of $t \in (0,1]$, ensuring stability in the limit $t \to 0^+$.
• To apply the result to the Kähler-Ricci flow on manifolds of intermediate Kodaira dimension $1 \leq \kod(X) \leq n-1$.
• To demonstrate that the $L^\infty$-bound depends only on the fibration $\pi$ and the $L^p$-norm of the density $F$.

Proposed method

• Rewrite the Monge-Ampère equation in the form $(\omega_t + dd^c \varphi_t)^n = f_t \omega_t^n$, where $f_t$ is a density uniformly bounded in $L^{p'}$ for some $p' > 1$.
• Establish uniform control of the density $f_t$ via Hölder’s inequality and the integrability of the Jacobian modulus $J_\pi^{-\alpha}$ for some $\alpha \in (0,1)$.
• Use the comparison principle between the measure $\mu_t = f_t \omega_t^n / \Vol_{\omega_t}(X)$ and the normalized capacity $\mathrm{Cap}_{\omega_t}/\Vol_{\omega_t}(X)$, showing $\mu_t(K) \leq C_0^n \left( \mathrm{Cap}_{\omega_t}(K)/\Vol_{\omega_t}(X) \right)^2$.
• Apply the Alexander-Taylor comparison theorem to relate the capacity of sublevel sets to the measure of the solution's sublevel sets.
• Use the comparison principle for $\omega_t$-plurisubharmonic functions to derive an a priori $L^\infty$-bound via the quantity $s_0(\omega_t)$, which is uniformly bounded in $t \in (0,1]$.
• Leverage the uniform boundedness of $\omega_t$-plurisubharmonic functions in $L^1(\omega_1^n)$ to show that $s_0(\omega_t)$ is uniformly bounded across $t$.

Experimental results

Research questions

• RQ1 Can uniform $L^\infty$-estimates be established for solutions to degenerate complex Monge-Ampère equations without geometric assumptions on the fibration $\pi: X \to Y$?
• RQ2 What is the dependence of the $L^\infty$-bound on the fibration $\pi$ and the $L^p$-norm of the density $F$?
• RQ3 How does the degeneration of the cohomology class to a non-big class affect the regularity of solutions to the Monge-Ampère equation?
• RQ4 Can the capacity comparison method be used to derive uniform bounds in the limit $t \to 0^+$ for families of Kähler forms $\omega_t = \pi^*\omega_Y + t\omega_X$?
• RQ5 What is the role of the Jacobian modulus $J_\pi$ in controlling the $L^p$-integrability of the density $f_t$?

Key findings

• The solutions $\varphi_t$ to the Monge-Ampère equation $(\star)_t$ satisfy a uniform $L^\infty$-bound: $\|\varphi_t\|_{L^\infty(X)} \leq M$ for all $t \in (0,1]$, where $M = M(\pi, \|F\|_p)$ is independent of $t$.
• The constant $M$ depends only on the fibration $\pi$ and the $L^p$-norm of $F$, not on any geometric assumptions on $\pi$.
• The density $f_t$ is uniformly bounded in $L^{p'}(\omega_t^n)$ for some $p' > 1$, with the bound depending only on $\pi$ and $\|F\|_p$.
• The measure $\mu_t = f_t \omega_t^n / \Vol_{\omega_t}(X)$ is uniformly strongly dominated by the normalized capacity $\mathrm{Cap}_{\omega_t}/\Vol_{\omega_t}(X)$, with a uniform constant $C_0^n$.
• The quantity $s_0(\omega_t)$, which controls the $L^\infty$-bound, is uniformly bounded in $t \in (0,1]$ by $e^n C_0^n (A + n)$, where $A$ depends only on $(X, \omega_1)$.
• The result implies that Theorems 1 and 2 in [KT] hold without the technical assumption on $\pi$, confirming a conjecture by Demailly and Pali.