Monge-Ampère Measure in Pluripotential Theory

• Monge-Ampère measure is a nonlinear operator in complex geometry that generates a positive Radon measure from bounded plurisubharmonic potentials on compact Hermitian or Kähler manifolds.
• It leverages Bedford–Taylor’s stability results to ensure that approximating sequences of potentials yield convergent measures, thus enabling rigorous analysis.
• The measure underpins the solvability and regularity of complex Monge-Ampère equations by controlling capacity-dominated and Hölder continuous potentials.

The Monge-Ampère measure is a nonlinear, highly degenerate elliptic operator central to pluripotential theory, complex geometry, and convex analysis. On a compact Kähler or Hermitian manifold, this measure associates to each bounded plurisubharmonic (psh) potential a positive Radon measure, encoding the local complex "volume" determined by the potential's curvature. The Monge-Ampère operator is essential in characterizing canonical geometric structures, determining the solvability of complex Monge-Ampère equations with singular right-hand sides, and in the regularity theory of nonlinear PDEs.

1. Foundational Definition and Main Properties

Let $(X, \omega)$ be a compact Hermitian manifold of complex dimension $n$, with $\omega$ a fixed smooth real (1,1)-form. A function $u: X \to \mathbb{R} \cup \{-\infty\}$ is called $\omega$-plurisubharmonic ($\omega$-psh) if $u$ is upper semicontinuous, not identically $-\infty$, and satisfies $\omega + dd^c u \geq 0$ in the sense of currents, where $d^c = \frac{i}{2\pi}(\bar{\partial}-\partial)$ and $dd^c = \frac{i}{\pi} \partial \bar{\partial}$.

For $u \in L^\infty(X) \cap \mathrm{PSH}(\omega)$, Bedford–Taylor's theory guarantees that the top wedge

$(\omega + dd^c u)^n$

is a well-defined nonnegative Radon measure on $X$. With normalization,

$\int_X (\omega + dd^c u)^n = \int_X \omega^n,$

so for the trivial potential $u \equiv 0$, the measure $(\omega + dd^c 0)^n = \omega^n$ gives the total volume form.

Continuity: If $u_j$ is a sequence of bounded $\omega$-psh functions converging uniformly to $u$, then $$(\omega + dd^c u_j)^n \rightarrow (\omega+dd^c u)^n$$ weakly as measures. The Monge-Ampère operator is thus stable under uniform convergence of bounded $\omega$-psh approximants.

2. Capacity, Domination, and Structural Classes of Measures

The global Monge-Ampère capacity is defined for any Borel set $E \subset X$ by

$$\mathrm{Cap}_\omega(E) = \sup \left\{ \int_E (\omega + dd^c \varphi)^n : \varphi \in \mathrm{PSH}(\omega),\ 0 \leq \varphi \leq 1 \right\}.$$

A measure $\mu$ is said to be dominated by capacity with weight $h$ (notation $\mu \in \mathcal{F}(X, h)$) if there exists $C_h>0$ such that

$$\mu(E) \leq C_h\, h(\mathrm{Cap}_\omega(E)),\quad \forall E \subset X,\ \text{Borel}.$$

Notable families include $\mathcal{H}(\tau)$ for $h(x) = C x^\tau$ and so-called moderate measures for $h(x) = C\,\exp(ax)$.

For admissible $h: (0, \infty)\to(0,\infty)$, the integrability condition

$\int_1^\infty \frac{1}{x[h(x)]^n} dx < +\infty$

is imposed, controlling the possible singularity of $\mu$ and ensuring that $\mu$ does not over-charge small capacity sets, which is critical for solvability and boundedness of solutions to the Monge-Ampère equation.

3. Existence and Regularity of Monge-Ampère Equations with Capacity-Dominated Measures

Main Existence Theorem (Kołodziej–Nguyen):

If $\mu \in \mathcal{F}(X,h)$ is a positive Radon measure with $\mu(X)>0$, then there exists a unique function $u \in \mathrm{PSH}(\omega) \cap C^0(X)$, normalized by $\sup_X u = 0$, and a constant $c>0$ such that

$$(\omega + dd^c u)^n = c\,\mu,\quad \int_X (\omega + dd^c u)^n = \int_X \omega^n.$$

Proof strategy:

• Approximation: One obtains a sequence of smooth measures $\mu_j \to \mu$, $\mu_j \in \mathcal{F}(X,h)$ uniformly, using mollifiers in local charts.
• Smooth Solvability: For each $\mu_j$, solve

$$(\omega + dd^c u_j)^n = c_j\,\mu_j,\quad \sup_X u_j = 0,$$

via Hermitian continuity method and Chern–Levine–Nirenberg estimates.

• Uniformity: Show $0 < \inf c_j \leq \sup c_j < \infty$ using local lower bounds for Monge-Ampère mass.
• Stability: Employ a capacity–mass stability estimate to guarantee $u_j \rightarrow u$ uniformly in $C^0$.
• Passage to the limit: $(\omega+dd^c u)^n = c\,\mu$ as the limit.

This analytic machinery ensures that the Monge-Ampère equation on compact Hermitian manifolds admits continuous quasi-psh solutions for measures with potentially large singular parts, provided those are controlled by capacity.

4. Hölder Potentials: Characterization via Measure Regularity

For finer regularity, such as Hölder continuity of the solution, a detailed relation is drawn between the regularity of $\mu$ (expressed via functionals) and the regularity of the solution.

Define

$$S = \left\{ v \in \mathrm{PSH}(\omega)\cap L^\infty(X): -1 \leq v \leq 0,\ \sup_X v = 0 \right\}.$$

A measure $\mu$ defines a functional

$$\mu\colon \mathrm{PSH}(\omega) \to \mathbb{R},\quad v \mapsto \int_X v\,d\mu.$$

$\mu$ is Hölder continuous on $S$ if, for some $\alpha>0$ and $C>0$, and all $u,v \in S$,

$|\mu(u) - \mu(v)| \leq C\,\|u-v\|_{L^1(X)}^\alpha.$

Theorem (Kołodziej–Nguyen, Dinh–Nguyen type):

$\mu \in \mathcal{H}(\tau)$ and $\mu$ is Hölder continuous on $S$ if and only if there is a Hölder continuous solution $u \in \mathrm{PSH}(\omega) \cap C^\beta(X)$, for some $\beta \in (0,1)$, with $(\omega + dd^c u)^n = c\,\mu$.

Key lemmas establish that local Hölder continuity of $\mu$ implies global regularity, and that domination of $\mu$ by the Monge-Ampère measure of a Hölder continuous potential ensures the functional is Hölder on $S$.

5. Connection to Kähler Case, Comparison with Previous Results

The capacity domination and Hölder regularity criteria generalize the classical results from the Kähler case (see Dinh–Nguyên (Dinh et al., 2012)) to the Hermitian setting. In the Kähler context, Hölder continuous super-potentials of measures are equivalent to the existence of Hölder continuous $\omega$-psh solutions to the Monge-Ampère equation. Conditions for Hölder solvability also include that $\mu$ admits a locally Hölder continuous potential, or belongs to function spaces such as Sobolev $W^{2n/p-2+\varepsilon,p}(X)$ (with $p>1$) or Besov spaces.

Locally, domination by Monge-Ampère measures of Hölder continuous potentials also characterizes measures admitting Hölder continuous solutions on compact Hermitian manifolds (Kolodziej et al., 2017).

6. Applications and Significance in Pluripotential Theory

These results extend the pluripotential theory of the complex Monge-Ampère operator on Kähler manifolds to the Hermitian setting, significantly enlarging the class of admissible right-hand-side measures. This includes measures with singularities as severe as those arising from capacity domination or as push-forwards under holomorphic dynamics, while still ensuring the solvability of the complex Monge-Ampère equation with continuous or Hölder continuous potentials.

The analytic and geometric flexibility provided by this theory underlies advances in complex geometry, including complex dynamics, the theory of canonical metrics (such as Hermitian–Einstein metrics), and the paper of singular Kähler–Einstein metrics. Measures dominated by capacity encompass all moderate measures, thus highlighing the breadth of applicability.

7. Schematic Summary of Key Classes

Measure class	Defining property	Solution existence/reg.
$\mathcal{F}(X,h)$	$\mu(E) \leq C_h\,h(\mathrm{Cap}_\omega(E))$	$u \in \mathrm{PSH}(\omega)\cap C^0$
$\mathcal{H}(\tau)$	$h(x) = Cx^\tau$	$u \in \mathrm{PSH}(\omega)\cap C^\beta$
Moderate measures	$h(x) = C\exp(ax)$	continuous/Hölder potential exists

The theory provides a precise bridge between analytic regularity of Monge-Ampère potentials and the fine measure-theoretic structure of $\mu$ expressed in terms of capacity bounds and Hölder continuity of induced functionals.