Variational property of the Monge-Ampère operator on Kähler manifolds (Q2780399)

Let \((V,g)\) be a compact Kähler manifold. For a real valued function \(\varphi\in C^{2,\alpha} (V)\) such that \(g'_{\lambda \overline\mu}= g_{\lambda \overline\mu} +\nabla_{\lambda \overline\mu} \varphi\) defines a positive definite form, set \(M(\varphi)= \det(g_{\lambda \overline\mu}+ \nabla_{\lambda \overline\mu} \varphi) \cdot\det(g_{\lambda \overline\mu})^{-1}\) -- the value of the Monge-Ampere operator on \(\varphi\). This is a function \(M: C^{2,\alpha}\to C^{0,\alpha}\). The authors write down explicitly a certain smooth functional \(F:C^{2,\alpha} \to\mathbb{R}\) such that its \(L^2\)-gradient is \(M\), i.e. NEWLINE\[NEWLINE\delta F(\varphi)h= \bigl\langle M(\varphi), h\rangle_{L_2}.NEWLINE\]NEWLINE This shows, in particular, that the Monge-Ampere equation is of variational type.