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Complex Monge-Ampère equation for positive $(p,p)$-forms on compact Kähler manifolds

Mathew George

TL;DR

This work develops a fully nonlinear complex Monge-Ampère-type theory for positive $(p,p)$-forms on compact Kähler manifolds, proving the existence and uniqueness (up to constants) of smooth solutions to $\mu_{\Omega_u}=e^{F+b}\omega^n$ with $\Omega_u=\Omega+i\partial\bar\partial u\wedge\omega^{p-1}$. The authors construct a robust analytic framework, establishing concavity and ellipticity of the associated operator, derive $C^0$, second-order, and gradient estimates via a mix of ABP, test-functions, and a Kronecker-product decomposition to control higher-order terms, and apply a continuity method to obtain existence. They also introduce a geometric flow for $(p,p)$-forms that preserves cohomology and generalizes the Kähler-Ricci flow, proving long-time existence and convergence (in a normalized form) to a canonical representative solving the elliptic equation. These results extend classical Calabi-Yau and Tosatti–Weinkove theories to higher $(p,p)$-forms and provide tools for canonical form selection within cohomology classes, with potential applications to geometric analysis and mathematical physics. The paper thus broadens the landscape of fully nonlinear PDEs on complex manifolds, offering new methods for higher-order form equations and associated flows.

Abstract

A complex Monge-Ampère equation for differential $(p,p)$-forms is introduced on compact Kähler manifolds. For any $1 \leq p < n$, we show the existence of smooth solutions unique up to adding constants. For $p=1$, this corresponds to the Calabi-Yau theorem proved by S. T. Yau, and for $p=n-1$, this gives the Monge-Ampère equation for $(n-1)$ plurisubharmonic functions studied by Tosatti-Weinkove. For other $p$ values, this defines a non-linear PDE that falls outside of the general framework of Caffarelli-Nirenberg-Spruck. Further, we define a geometric flow for higher-order forms that preserves their cohomology classes, and extends the Kähler-Ricci flow naturally to $(p,p)$-forms. As a consequence of our main theorem, we show that this flow exists in a maximal time interval and can be shown to converge under some assumptions. A modified flow is introduced and the convergence of the associated normalized flow is shown.

Complex Monge-Ampère equation for positive $(p,p)$-forms on compact Kähler manifolds

TL;DR

This work develops a fully nonlinear complex Monge-Ampère-type theory for positive -forms on compact Kähler manifolds, proving the existence and uniqueness (up to constants) of smooth solutions to with . The authors construct a robust analytic framework, establishing concavity and ellipticity of the associated operator, derive , second-order, and gradient estimates via a mix of ABP, test-functions, and a Kronecker-product decomposition to control higher-order terms, and apply a continuity method to obtain existence. They also introduce a geometric flow for -forms that preserves cohomology and generalizes the Kähler-Ricci flow, proving long-time existence and convergence (in a normalized form) to a canonical representative solving the elliptic equation. These results extend classical Calabi-Yau and Tosatti–Weinkove theories to higher -forms and provide tools for canonical form selection within cohomology classes, with potential applications to geometric analysis and mathematical physics. The paper thus broadens the landscape of fully nonlinear PDEs on complex manifolds, offering new methods for higher-order form equations and associated flows.

Abstract

A complex Monge-Ampère equation for differential -forms is introduced on compact Kähler manifolds. For any , we show the existence of smooth solutions unique up to adding constants. For , this corresponds to the Calabi-Yau theorem proved by S. T. Yau, and for , this gives the Monge-Ampère equation for plurisubharmonic functions studied by Tosatti-Weinkove. For other values, this defines a non-linear PDE that falls outside of the general framework of Caffarelli-Nirenberg-Spruck. Further, we define a geometric flow for higher-order forms that preserves their cohomology classes, and extends the Kähler-Ricci flow naturally to -forms. As a consequence of our main theorem, we show that this flow exists in a maximal time interval and can be shown to converge under some assumptions. A modified flow is introduced and the convergence of the associated normalized flow is shown.

Paper Structure

This paper contains 17 sections, 17 theorems, 303 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matrix associated to a $(p,p)$-form:
  4. Notations and Conventions:
  5. Volume form of a positive $(p,p)$-form:
  6. Curvature and connection:
  7. Concavity of the operator:
  8. Ellipticity of the operator $G$:
  9. $C^0$ estimates
  10. Second-order estimates
  11. Kronecker product structure of $G^{i \bar{l}, k \bar{j}}$
  12. The term $Z^{I \bar{J}} Z^{K \bar{L}} \nabla_{1} Z_{L \bar{I}} \nabla_{\bar{1}} Z_{J \bar{K}}$
  13. The term $\dfrac{1}{\lambda_1^2} \sigma(b,l)Z^{I'_b \bar{I}'_l}\lambda_{1, l}\lambda_{1, \bar{b}}$
  14. Gradient estimates and existence of solutions
  15. The Geometric flow
  16. ...and 2 more sections

Key Result

Theorem 1.1

Given a compact Kähler manifold $(M, \omega)$, a smooth function $F$ on $M$, and a positive $(p,p)$-form $\Omega$, there exists a unique smooth function $u$ with $\sup\limits_M u = 0$, and a unique constant $b$ such that where $\Omega_u = \Omega + \sqrt{-1} \partial \bar{\partial} u \wedge \omega^{p-1}$ is a positive form.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Proposition 2.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 28 more