---
title: Generalized Monge-Ampère Equation
url: https://www.emergentmind.com/topics/generalized-monge-ampere-equation
type: topic
---

# Generalized Monge-Ampère Equation

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The generalized Monge–Ampère equation is not a single canonical PDE but a family of fully nonlinear equations and exterior differential systems that extend the classical Monge–Ampère paradigm by coupling top-degree determinants or wedge powers to lower-degree terms, mixed forms, or minor conditions on jet spaces. In complex geometry, one encounters equations of the form
$$
\omega_\varphi^n-\sum_{k=1}^{n-1}c_k\,\chi^{\,n-k}\wedge\omega_\varphi^k=f\,\chi^n,
$$
weak-current identities involving nonpluripolar products,
$$
\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1}c_k\langle\chi_\psi^k\wedge\omega^{n-k}\rangle,
$$
and determinant relations for generalized Kähler potentials; in jet-space formulations, generalized Monge–Ampère equations are systems built from minors of prolongation matrices and are equivalent to special exterior differential systems [2006.01530] [2605.29258] [1005.5658] [2008.10203].

## 1. Families of equations and representative forms

A broad Kähler formulation appears in Pingali’s equation
$$
a_0\wedge(\omega+i\partial\bar\partial\varphi)^n+a_1\wedge(\omega+i\partial\bar\partial\varphi)^{n-1}+\cdots+a_{n-1}\wedge(\omega+i\partial\bar\partial\varphi)=n\,\Omega,
$$
with $\int_X\varphi\,\omega^n=0$ on a compact Kähler manifold [1205.1266]. In dimension three, Pingali also studies the torus equation
$$
(\omega+dd^c\phi)^3-\alpha\wedge(\omega+dd^c\phi)
=
e^F\,(\omega^3-\alpha\wedge\omega),
$$
for a flat complex $3$-torus and a constant-coefficient $(2,2)$-form $\alpha$ satisfying $\alpha-\omega^2>0$ and $\omega^3-\alpha\wedge\omega>0$ [1310.1656].

On projective and Hermitian manifolds, generalized complex Monge–Ampère equations are written as mixed wedge equations. Datar–Pingali study
$$
\omega_\varphi^n-\sum_{k=1}^{n-1}c_k\,\chi^{\,n-k}\wedge\omega_\varphi^k=f\,\chi^n,
$$
with a cone condition expressing admissibility, while Sun considers
$$
\sum_{a=1}^n c_a\,\chi_u^{\,n-a}\wedge\omega^a=\psi\,\omega^n,
\qquad \chi_u>0,\quad \sup_Mu=0,
$$
on closed Hermitian manifolds [2006.01530] [1412.8192].

The terminology also extends beyond scalar potentials. For forms of positive degree, the original Monge–Ampère-type equation of Dinew–Popovici seeks a real $(m-1,m-1)$-form $u$ satisfying
$$
[\,\star_\omega((\alpha+i\,u)\wedge\omega_{n-m-1})\,]^n=dV,
$$
supplemented by a gauge condition, while generalized Kähler geometry yields a determinant identity for a single real generalized Kähler potential $K$,
$$
\det\!\begin{pmatrix}
-\,K_{L\bar L} & -\,K_{LR}\\
-\,K_{\bar R\bar L} & -\,K_{\bar R R}
\end{pmatrix}
=
\det\!\begin{pmatrix}
K_{c\bar c} & K_{c\bar t}\\
K_{t\bar c} & K_{t\bar t}
\end{pmatrix}.
$$
These formulations show that the terminology is structurally rather than syntactically uniform: what remains common is a fully nonlinear positivity-preserving relation generalizing the Monge–Ampère determinant [2606.23981] [1005.5658].

## 2. Positivity, cohomological balance, and solvable regions

A recurrent feature is a cohomological or volume-normalization constraint. In Murakami’s Kähler setup, $(X^n,\omega)$ is a compact Kähler manifold, $[\chi]$ is a nef real $(1,1)$ class, the constants $c_1,\dots,c_{n-1}$ are nonnegative, and $c_0$ is continuous, subject to
$$
\int_X [\chi]^n-\sum_{k=1}^{n-1} c_k\,[\chi]^k\wedge[\omega]^{n-k}
=
\int_X c_0\,\omega^n.
$$
In Pingali’s torus equation one requires
$$
\int_X e^F\,(\omega^3-\alpha\wedge\omega)=\int_X(\omega^3-\alpha\wedge\omega),
$$
and on a closed almost Kähler surface the analogue is
$$
\int_M e^f\,\omega^2=\int_M\omega^2.
$$
These balance conditions ensure compatibility between the nonlinear equation and the fixed background cohomology or volume form [2605.29258] [1310.1656] [2412.18361].

Admissibility is encoded by positivity cones and subsolution conditions. In Murakami’s interior region, the generalized Monge–Ampère equation has a classical $C^4$-solution precisely when there exists a smooth $\mathbf C$-subsolution
$$
n\,\chi'^{\,n-1}-\sum_{k=1}^{n-1}c_k\,k\,\chi'^{\,k-1}\wedge\omega^{n-k}>0.
$$
The boundary case is obtained by approximating $[\chi]$, $\omega$, and the coefficients $c_k$ by nearby data for which the relevant positivity tests hold strictly [2605.29258].

For projective manifolds, Datar–Pingali prove an exact numerical criterion. Solvability of the mixed equation, existence of a Kähler metric in the class satisfying the pointwise cone condition, and positivity of the intersection numbers
$$
\int_V \Bigl(\omega^{n-p}-\sum_{k=1}^{n-1}c_k\,\omega^{\,n-p-k}\wedge\chi^k\Bigr)>0
$$
for every irreducible subvariety $V\subset M$ of codimension $p\ge1$ are equivalent [2006.01530]. In this form, solvability is characterized by a Nakai–Moishezon-type criterion.

## 3. Weak, degenerate, and singular solutions

Murakami develops a weak-current theory for boundary classes using nonpluripolar products. If $\psi\in\operatorname{PSH}(X,\chi)$ with $\sup\psi=0$, then $\chi_\psi=\chi+i\partial\bar\partial\psi$ is a closed positive current, and $\psi$ is called a weak solution when
$$
\langle\chi_\psi^n\rangle=\sum_{k=0}^{n-1} c_k\,\langle\chi_\psi^k\wedge\omega^{n-k}\rangle
$$
together with the positivity mass-inequalities for every $p=1,\dots,n-1$. Under the boundary-case assumption, there exists a unique $\chi$-psh weak solution, and the associated mixed Hessian flow converges weakly as currents to that solution [2605.29258].

On compact Hermitian manifolds, Nguyen proves existence and uniqueness of continuous solutions for exponential and degenerate equations. For $f\in L^p$, $p>1$, one obtains a unique $\varphi\in\operatorname{PSH}(\omega)\cap C^0(X)$ with $\sup_X\varphi=0$ solving
$$
(\omega+dd^c\varphi)^n=e^\varphi f\,\omega^n,
$$
and likewise a unique continuous solution for semi-positive closed backgrounds $\beta\ge0$ with $\int_X\beta^n>0$ [1501.00891]. Li–Lin–Zhou develop a complementary Hermitian theory for
$$
(\beta+dd^c u)^n=e^{\lambda u}\mu
$$
in the relative full-mass class $\mathcal E(X,\beta,\phi)$, where $\mu$ is a positive non-pluripolar Radon measure, $\lambda\ge0$, and $\phi$ is a model potential; uniqueness follows from the comparison principle, and one moreover has
$$
\nu(u,x)=\nu(\phi,x)
\quad\text{for every }x\in X,
$$
so the solution inherits the Lelong-number profile of the model potential [2311.14958].

Di Nezza–Lu treat singular right-hand sides on quasi-projective varieties. For
$$
(\omega+dd^c\varphi)^n=f\,\omega^n
\quad\text{on }X\setminus D,
$$
with $f$ smooth outside a divisor and satisfying the conditions $H_f$ or, in the SNC setting, $S(B,\alpha)$, there exists a unique normalized $\varphi\in\mathcal E(X,\omega)$ that is smooth on $X\setminus D$. The asymptotics depend sharply on $\alpha$: if $\alpha>1$, $\varphi$ extends continuously; if $\alpha=1$, one gets double-logarithmic lower bounds; and if $0<\alpha<1$, the solution grows at a power rate in $-\log|s_j|$ [1401.6398].

## 4. Analytic frameworks and proof strategies

Method-of-continuity arguments dominate the smooth theory. Pingali’s torus theorem uses openness via positivity of the linearized operator and closedness from uniform $C^0$, $C^1$, and $C^{1,1}$ estimates, followed by Evans–Krylov and Schauder regularity. The $C^1$ estimate uses Blocki’s device
$$
B=\log(1+|\nabla\phi|^2)-A\phi,
$$
and the $C^{1,1}$ estimate applies the maximum principle to $\operatorname{tr}_\omega(\omega_\phi)$, with the mixed term $\alpha\wedge\omega_\phi$ controlled by convexity [1310.1656]. In the positive-degree setting, the originally proposed gauge $\,\Lambda^{m-2}\Delta u=0\,$ is shown to be analytically too rigid, and the replacement is the $(a,b)$-gauge
$$
(m-1)\Lambda^{m-2}\Delta u=(1+a)\,i\,\Lambda^{m-1}u+(1+b)\Lambda^{m-1}\Delta u\cdot\omega,
$$
which leads to a scalar $(a,b)$–Monge–Ampère equation with $C^0$, gradient, and $C^2$ estimates, and then Evans–Krylov–Schauder bootstrapping [2606.23981].

On closed Hermitian manifolds, Sun derives the a priori estimates needed for admissible solutions of
$$
\sum_{a=1}^n c_a\,\chi_u^{\,n-a}\wedge\omega^a=\psi\,\omega^n.
$$
The $C^0$ bound comes from Moser iteration, the partial $C^2$ bound from a maximum principle applied to $\log\operatorname{tr}_\omega(\chi_u)+\phi(u)$, and the improved gradient estimate from a Blocki–Phong–Sturm–Guan–Sun argument [1412.8192]. The parabolic counterpart evolves by
$$
\partial_t u=\log\frac{\chi_u^n}{\sum_{a=1}^n b_a\,\chi_u^{\,n-a}\wedge\omega^a}-\log\psi(x),
$$
and under the cone condition the normalized flow has uniform $C^\infty$ estimates and converges in $C^\infty$ to the elliptic solution [1501.04255].

Murakami’s boundary-case theory combines viscosity and pluripotential methods. The limit of smooth approximants is first shown, via the CIL lemma and convolution, to be a viscosity subsolution; a pluripotential mass-comparison in the spirit of Bedford–Taylor and BEGZ upgrades the inequality to equality; and uniqueness follows from a viscosity/pluripotential maximum principle together with convexity of the operator and mass conservation [2605.29258]. In the projective setting, Datar–Pingali introduce a degenerate concentration of mass argument, producing weak limit currents with positive mass on ample divisors and then building barrier currents by resolution, induction on dimension, and Blocki–Kołodziej/Richberg gluing [2006.01530].

## 5. Geometric and physical applications

Several generalized Monge–Ampère equations are motivated by geometric PDE problems. Murakami treats the generalized Monge–Ampère equation and the supercritical deformed Hermitian–Yang–Mills equation simultaneously; under the same boundary hypotheses, the mixed Hessian flow converges weakly to the unique weak generalized Monge–Ampère solution, and the dHYM flow satisfies
$$
\|Q(\alpha_t)+\cot\theta\|_{L^2(\omega)}\to0,
$$
hence converges in currents to the weak dHYM solution [2605.29258].

Pingali’s flat-torus equation has a vector-bundle application. If $h=e^{-2\phi}h_0$ is a conformal change of a Hermitian metric on a holomorphic vector bundle, then
$$
F_h=F_0+dd^c\phi\otimes\operatorname{Id}_V,
$$
and in complex dimension three the condition $\operatorname{tr}(F_h^3)=\eta$ becomes exactly the generalized Monge–Ampère equation after identifying
$$
\alpha=\frac{k-1}{k}\operatorname{tr}(F_0^2),
\qquad
\omega^3-\alpha\wedge\omega=\frac1k\operatorname{tr}(F_0^3).
$$
Thus prescribed top Chern character forms can be realized by the Chern connection of a conformally modified metric under the stated positivity hypotheses [1310.1656].

For forms of positive degree, the $(a,b)$ framework yields the qualitative part of Demailly’s transcendental Morse inequality unconditionally. When $b=0$, the equation reduces to the classical complex Monge–Ampère equation, and Yau’s theorem produces a strictly positive Kähler current in the Bott–Chern class $[\alpha]-[\beta]$ under the integral positivity hypothesis stated in the paper [2606.23981].

In generalized Kähler geometry, the determinant equation for the generalized Kähler potential is exactly the local condition for a generalized Kähler manifold to be a generalized Calabi–Yau metric. Its solutions yield target-space fields $(g,H,\phi)$ solving the Type II supergravity equations with zero Ramond–Ramond fluxes; in the Kähler limit it reduces to the standard complex Monge–Ampère equation $\det(K_{i\bar j})=\mathrm{const}$ [1005.5658]. On closed almost Kähler surfaces, the equation
$$
(\omega+\mathcal D_J^+(\varphi))^2=e^f\,\omega^2
$$
has a unique smooth solution with full $C^{k+2,\alpha}$ estimates, and this is used to verify Donaldson’s conjecture for tamed almost complex $4$-manifolds under the hypothesis $h_J^-=b^+-1$ [2412.18361].

## 6. Jet-space and exterior-differential-system generalizations

A distinct line of work interprets generalized Monge–Ampère equations as special exterior differential systems on jet spaces. On the $k$-jet space $J^k(n,m)$ with canonical contact system $C_k$, Kawamata–Shibuya define a generalized Monge–Ampère system
$$
I=\{\,C_k,Y_1,\dots,Y_r\,\}_{\mathrm{diff}}
$$
generated by the contact forms and a finite collection of differential forms $Y_u$. The associated generalized Monge–Ampère equation is obtained from linear combinations of minors of the prolongation matrix $M(n,m;k)$, and Theorem 1.1 states that integral manifolds of the GMAS correspond bijectively to solutions of the GMAE; conversely, every such GMAE arises from a uniquely determined GMAS [2008.10203].

This framework recovers the classical second-order Monge–Ampère equation when $(n,m,k,r,\ell)=(2,1,1,1,2)$, and it also accommodates examples not usually classified under the Monge–Ampère label. Kawamata–Shibuya verify that the Korteweg–de Vries equation
$$
u_t+2u\,u_x+u_{xxx}=0
$$
and the Cauchy–Riemann system
$$
u_x-v_y=0,\qquad u_y+v_x=0
$$
are both generalized Monge–Ampère equations in this sense [2008.10203].

Kawamata’s later study of second-order generalized Monge–Ampère equations on the plane encodes systems such as
$$
A\,z_{xx}+B\,z_{xy}+C=0,\qquad
A\,z_{xy}+B\,z_{yy}+D=0
$$
as generalized Monge–Ampère systems on $J^1(\mathbb R^2,\mathbb R)$, reduces them via Cauchy characteristics, and constructs geometric singular solutions. The resulting wave fronts are classified by standard criteria into cuspidal edge, swallowtail, butterfly, and, in the non-generic case, beaks singularities [2111.06695]. A plausible implication is that the expression “generalized Monge–Ampère equation” now names a mathematically coherent but genuinely plural class of nonlinear structures, unified less by a single formula than by determinant-like nonlinearity, positivity cones, and geometric origin.

Source: https://www.emergentmind.com/topics/generalized-monge-ampere-equation