The Monge-Ampère equation on compact Kähler manifolds. (English) Zbl 1039.32050

Kołodziej, Sławomir

Indiana Univ. Math. J. 52, No. 3, 667-686 (2003).

Let \(M\) be a compact \(n\)-dimensional Kähler manifold with the fundamental form \(\omega\) given in local coordinates as \(\omega=\frac{i}2 \sum_{j,k=1}^n g_{j,\overline{k}} \,dz^j\wedge d\overline z^k\) and normalized by the condition \(\int_M\omega^n=1\). Let \(\mathcal{PSH}(\omega)\) be the set of all \(\omega\)-plurisubharmonic functions, i.e. those continuous functions \(\varphi\) on \(M\) for which \(\omega+dd^c\varphi\geq0\). For a Borel set \(E\subset M\) define cap\({}_\omega(E):=\sup\{\int_E (\omega+dd^c\varphi)^n: \varphi\in\mathcal{PSH}(\omega),\;0\leq\varphi\leq1\}\). Let \(F:\mathbb R_+\rightarrow\mathbb R_+\), \(F(x)=\frac{Ax}{h(x^{-1/n})}\), where \(A>0\) and \(h:\mathbb R_+\rightarrow[1,+\infty)\) is a non-decreasing function with \(\int_1^{+\infty} \frac{dy}{yh^{1/n} (y)}<+\infty\). Define \[ \mathcal F(F):=\{f\in L^1(M): f\geq 0, \int_Mf\omega^n=1, \int_Ef\omega^n\leq F(\text{cap}_\omega (E))\text{ for any Borel set }E\subset M\}. \] The main result of the paper is the following stability theorem. Assume that \(1\in\mathcal F(F)\) and let \(f,g\in\mathcal F(F)\). Suppose that \(\varphi, \psi\in\mathcal{PSH}(\omega)\) are solutions of the equations \((\omega+dd^c\varphi)^n= f\omega^n\), \((\omega+dd^c\psi)^n= g\omega^n\) with \(\max_M(\varphi-\psi)= \max_M(\psi-\varphi)\). Then there exist constants \(a=a(F)\), \(c=c(n)>0\) such that if we put \(\kappa_F(s):=c(n)A^{1/n} (1+a)(\int_{s^{-1/n}}^{+\infty} \frac{dy}{yh^{1/n}(y)}+ h^{-1/n}(s^{-1/n}))\), \(\gamma(t):=\frac {(2a)^n}{(a+1)^n} \frac{(\frac32)^{1/n}-1}{3} \kappa_F^{-1}(t)\), then the inequality \(\| f-g\| _1\leq\gamma(t)t^{n+3}\) implies that \(\| \varphi-\psi\| _\infty\leq(4a+2)t\) for \(t<t_0\) with \(t_0\) depending only on \(\gamma\).
The author presents also applications of the above stability theorem to the uniqueness and equicontinuity of solutions of the complex Monge-Ampère equation.

Reviewer: Marek Jarnicki (Kraków)

MSC:

32W20	Complex Monge-Ampère operators
53C55	Global differential geometry of Hermitian and Kählerian manifolds
32U40	Currents
32U05	Plurisubharmonic functions and generalizations