[Paper Review] Complex Monge-Ampère equation for measures supported on real submanifolds

Duc‐Viet Vu|arXiv (Cornell University)|Aug 9, 2016

Geometry and complex manifolds13 references4 citations

TL;DR

This paper establishes the existence of a Hölder continuous solution to the complex Monge-Ampère equation $(dd^c\varphi + \omega)^n = \mu$ when the measure $\mu$ is supported on a generic $\mathcal{C}^3$ real submanifold $K$ of a compact Kähler manifold $X$, provided $\mu$ has an $L^p$ density for $p > 1$. The key contribution is proving that such measures have Hölder continuous super-potentials, which implies Hölder regularity of the solution with an explicit Hölder exponent depending on $p$, $d$, and $n$.

ABSTRACT

Let $(X,ω)$ be a compact $n$-dimensional Kähler manifold on which the integral of $ω^n$ is $1$. Let $K$ be an immersed real $\mathcal{C}^3$ submanifold of $X$ such that the tangent space at any point of $K$ is not contained in any complex hyperplane of the (real) tangent space at that point of $X.$ Let $μ$ be a probability measure compactly supported on $K$ with $L^p$ density for some $p>1.$ We prove that the complex Monge-Ampère equation $(dd^c φ+ ω)^n=μ$ has a Hölder continuous solution.

Motivation & Objective

• Address the regularity of solutions to the complex Monge-Ampère equation $(dd^c\varphi + \omega)^n = \mu$ when $\mu$ is supported on a generic real submanifold.
• Identify a class of measures—specifically, probability measures with $L^p$ densities supported on generic $\mathcal{C}^3$ submanifolds—for which the solution is Hölder continuous.
• Establish that the super-potential of such measures is Hölder continuous, which implies Hölder regularity of the solution via known stability results.
• Provide an explicit Hölder exponent for the solution in terms of $p$, the real codimension $d$, and the complex dimension $n$ of the ambient manifold.
• Extend the known class of measures yielding Hölder continuous solutions beyond those with densities relative to volume forms.

Proposed method

• The proof relies on constructing a family of analytic discs partly attached to the generic submanifold $K$, using a modified version of the solution to a Beltrami-type equation with a parameter $t$.
• Analytic discs are constructed via harmonic extensions and a perturbation of a reference function $u_0$ with controlled properties, ensuring the discs remain close to $K$ near the boundary.
• The construction ensures the Jacobian determinant of the disc map satisfies a lower bound proportional to $(1 - |z|)^{d-1}$, which controls the distortion and enables measure estimates.
• Super-potential regularity is established by estimating the $L^1$-distance between $\omega$-p.s.h. functions and using the geometry of the analytic discs to bound the super-potential variation.
• The Hölder continuity of the super-potential is derived from the uniform control of the measure of the image of the discs under the solution map.
• Finally, the result is combined with known stability and regularity theorems (e.g., from Demailly, Dinew, Guedj, etc.) to deduce Hölder continuity of the solution $\varphi$.

Experimental results

Research questions

• RQ1 What is the optimal Hölder regularity of the solution to the complex Monge-Ampère equation when the measure $\mu$ is supported on a generic real submanifold with $L^p$ density?
• RQ2 Can the super-potential of a measure supported on a generic $\mathcal{C}^3$ submanifold be shown to be Hölder continuous, and if so, with what exponent?
• RQ3 How does the geometry of analytic discs partly attached to a generic submanifold contribute to estimating super-potentials and proving regularity?
• RQ4 Does the class of measures yielding Hölder continuous solutions to the complex Monge-Ampère equation include those supported on submanifolds of positive codimension with $L^p$ densities?
• RQ5 Can the Hölder exponent of the solution be explicitly quantified in terms of the codimension $d$, the dimension $n$, and the integrability exponent $p$ of the density?

Key findings

• The complex Monge-Ampère equation $(dd^c\varphi + \omega)^n = \mu$ has a unique $\omega$-p.s.h. solution $\varphi$ that is Hölder continuous with exponent $\alpha < \frac{2(p-1)}{3d(n+1)p}$, where $p > 1$ is the integrability exponent of the density of $\mu$.
• The super-potential of the measure $\mu = \mathbf{1}_{\tilde{K}} \cdot \mathrm{vol}_K$ is Hölder continuous with exponent $\alpha < \frac{1}{3d}$, where $\tilde{K}$ is a compact subset of the generic $\mathcal{C}^3$ submanifold $K$.
• Corollary 1.2 establishes an exponential integrability estimate: there exist constants $\alpha > 0$ and $c > 0$ such that $\int_{\tilde{K}} e^{-\alpha \varphi} \, d\mathrm{vol}_K \leq c$ for all $\omega$-p.s.h. functions $\varphi$ with $\sup_X \varphi = 0$.
• The proof remains valid if $K$ is $\mathcal{C}^{2,\beta}$ for some $\beta \in (0,1)$, with the Hölder exponent adjusted accordingly by replacing $\mathcal{C}^{2,1/2}$ regularity with $\mathcal{C}^{2,\beta'}$ for $\beta' < \beta$.
• The result extends the class of measures for which Hölder continuous solutions exist beyond those with densities relative to the ambient volume form, including measures supported on submanifolds of positive codimension.
• The construction of analytic discs partly attached to $K$ provides a geometric mechanism to control the super-potential and enables the derivation of the Hölder estimate via measure-theoretic and potential-theoretic arguments.