[Paper Review] Hölder continuity of solutions to the complex Monge-Ampère equation with the right hand side in L^p. The case of compact Kähler manifolds

Sławomir Kołodziej|ArXiv.org|Nov 2, 2006

Geometry and complex manifolds4 references4 citations

TL;DR

This paper establishes the Hölder continuity of solutions to the complex Monge-Ampère equation $(\omega + dd^c u)^n = f\omega^n$ on compact Kähler manifolds when the right-hand side $f$ belongs to $L^p(M)$ for $p > 1$. Using regularization techniques, stability estimates, and a three circles-type inequality, the authors prove that the solution $u$ is Hölder continuous with an exponent depending on $p$, the manifold $M$, and $\|f\|_p$, extending previous results on continuity to higher regularity.

ABSTRACT

We prove that on compact Kähler manifolds solutions to the complex Monge-Ampère equation, with the the right hand side in $L^p, p>1,$ are Hölder continuous.

Motivation & Objective

• To establish Hölder continuity of $\omega$-plurisubharmonic solutions to the complex Monge-Ampère equation on compact Kähler manifolds when the right-hand side $f$ is in $L^p(M)$ for $p > 1$.
• To extend the known continuity results for $f \in L^p$, $p > 1$, to Hölder continuity, providing a quantitative modulus of continuity.
• To adapt techniques from strictly pseudoconvex domains, particularly the regularization and stability approach of [GKZ], to the compact Kähler setting.
• To address the regularity of potentials arising in the Kähler-Ricci flow, where $f$ may blow up along a subvariety but remain in $L^p$ for $p > 1$.

Proposed method

• Regularize the solution $u$ via local averaging: $u_{j,\delta}(z) = \max_{|w|<\delta} u(z+w)$ in coordinate charts.
• Define auxiliary functions $\chi(\delta) = \delta^{-\alpha} \max_j \max_z (u_{j,\delta} - u)(z)$ and $\eta(\delta) = \max_j \max_z (u_{j,C\delta} - u_{j,\delta})(z)$ to track oscillation decay.
• Use a comparison principle and stability estimate from [K2] to bound $\|u - v\|_\infty$ in terms of $\|f - g\|_1^{1/(n+3+\epsilon)}$ for normalized solutions.
• Apply the three circles theorem in the form $u_{j,N\delta} - u_{j,\delta} \geq \frac{\log N}{\log C}(u_{j,C\delta} - u_{j,\delta})$ to relate oscillations at different scales.
• Construct a subsolution $u_\delta$ by combining regularized $u_{j,\delta}$ with cutoff functions $\rho_j$, ensuring $dd^c u_\delta + \omega > 0$.
• Derive a contradiction by assuming $\chi(\delta) > \max(9, \chi(N\delta))$, leading to a set $E$ with small measure where $f$ is effectively zero, and use the comparison principle to violate the Monge-Ampère measure.

Experimental results

Research questions

• RQ1 Does the solution $u$ to the complex Monge-Ampère equation $(\omega + dd^c u)^n = f\omega^n$ on a compact Kähler manifold become Hölder continuous when $f \in L^p(M)$ for $p > 1$?
• RQ2 Can the stability estimate from [K2], which controls $\|u - v\|_\infty$ via $\|f - g\|_1^{1/(n+3+\epsilon)}$, be used to upgrade continuity to Hölder continuity?
• RQ3 How does the Hölder exponent depend on the $L^p$ norm of $f$, the manifold $M$, and the dimension $n$?
• RQ4 Can the regularization and oscillation decay techniques used in strictly pseudoconvex domains be adapted to the compact Kähler setting?
• RQ5 What is the regularity of the potential in the Kähler-Ricci flow when the limit metric has singularities along a subvariety but $f$ remains in $L^p$?

Key findings

• The solution $u$ to the complex Monge-Ampère equation is Hölder continuous whenever $f \in L^p(M)$ for $p > 1$, with the Hölder exponent depending on $p$, $M$, and $\|f\|_p$.
• The Hölder exponent $\alpha$ satisfies $\alpha < \frac{1}{q(n+3+\epsilon)+1}$, where $q$ is the H"older conjugate of $p$, and $\epsilon > 0$ is arbitrary.
• The proof relies on a contradiction argument assuming $\chi(\delta) > \max(9, \chi(N\delta))$, which leads to a set $E$ of small measure where $f$ is effectively zero.
• The measure of $E$ is bounded by $c_4 \delta^{1-\alpha}$, and Hölder's inequality yields $\int_E f \omega^n \leq c_5 \delta^{(1-\alpha)/q}$, which controls the $L^1$-difference in the stability estimate.
• The contradiction arises from the comparison principle: $\int_U (dd^c u_\delta + \omega)^n > 0$ but $\int_U (dd^c v + \omega)^n = \int_E g \omega^n = 0$, violating the positivity of the Monge-Ampère measure.
• The result extends to compact Kähler orbifolds and can be applied to potentials arising in the Kähler-Ricci flow on projective manifolds with singular limits in $L^p$.