[Paper Review] Solutions to degenerate complex Hessian equations

Lu Hoang Chinh|arXiv (Cornell University)|Feb 11, 2012

Geometry and complex manifolds32 references3 citations

TL;DR

This paper establishes the existence and uniqueness of continuous solutions to degenerate complex Hessian equations on compact Kähler manifolds, using a potential theory framework based on $(\omega,m)$-subharmonic functions and capacity estimates. For rational homogeneous manifolds with invariant Kähler forms, the solution is shown to be H"older continuous with an explicit exponent depending on dimension and $m$. The results generalize both Laplace and Monge-Ampflre equations in the complex setting.

ABSTRACT

Let $(X,ω)$ be an $n$-dimensional compact Kähler manifold. We study degenerate complex Hessian equations of the form $(ω+dd^cφ)^m\wedge ω^{n-m}=F(x,φ)ω^n.$ Under some natural conditions on $F$, this equation has a unique continuous solution. When $(X,ω)$ is rational homogeneous we further show that the solution is Hölder continuous.

Motivation & Objective

• To establish a potential theory for degenerate complex Hessian equations on compact K"ahler manifolds.
• To prove existence and uniqueness of continuous solutions under natural integrability and monotonicity conditions on the density $F(x,\varphi)$.
• To extend the theory to rational homogeneous manifolds where higher regularity (H"older continuity) of solutions is achieved.
• To generalize the complex Monge-Ampflre and Laplace equations within a unified Hessian framework.

Proposed method

• Introduce the class of $(\omega,m)$-subharmonic functions as a generalization of $\omega$-plurisubharmonic functions.
• Define the complex Hessian operator on bounded, quasi-continuous $(\omega,m)$-subharmonic functions using a capacity-based quasi-uniform convergence concept.
• Establish a comparison principle and continuity of the Hessian operator under quasi-uniform convergence.
• Use a fixed-point argument in a complete metric space of continuous functions to prove existence of solutions.
• Leverage group-invariant regularization via Haar integration on rational homogeneous spaces to improve regularity.
• Apply $L^2$ and $L^\infty$ estimates with capacity techniques to derive H"older continuity of the solution.

Experimental results

Research questions

• RQ1 Under what conditions does the degenerate complex Hessian equation $(\omega + dd^c\varphi)^m \wedge \omega^{n-m} = F(x,\varphi)\omega^n$ admit a continuous solution on a compact K"ahler manifold?
• RQ2 How can a consistent potential theory be developed for the complex Hessian operator when regularization of subharmonic functions fails?
• RQ3 What additional geometric structure (e.g., rational homogeneous space) enables the improvement of continuity to H"older regularity?
• RQ4 Can the solution be uniquely determined up to an additive constant under natural monotonicity and integrability conditions on $F$?

Key findings

• A unique continuous solution exists to the degenerate complex Hessian equation under conditions (F1)-(F3), unique up to an additive constant.
• When $F(x,\cdot)$ is strictly increasing in $\varphi$, the solution is unique without additive ambiguity.
• On rational homogeneous manifolds with $K$-invariant $\omega$, the solution is H"older continuous with exponent $\gamma < \frac{2(mp - n)}{mnp + 2mp - 2n}$.
• The H"older exponent reduces to the known exponent from [EGZ09] when $m = n$, recovering the Monge-Ampflre case.
• The solution class is stable under group averaging, enabling regularization and regularity transfer in symmetric spaces.
• The capacity-based framework ensures continuity of the Hessian operator under quasi-uniform convergence, extending Bedford-Taylor theory to the Hessian setting.