[Paper Review] The parabolic flows for complex quotient equations

Wei Sun|arXiv (Cornell University)|Dec 3, 2017

Geometry and complex manifolds26 references3 citations

TL;DR

This paper establishes the long-time existence and smooth convergence of a parabolic flow for complex quotient equations on closed Kähler manifolds. By applying the parabolic flow method with a $χ$-subsolution condition and uniform lower bound on the right-hand side $ψ$, it proves convergence to a solution of the complex quotient equation, thereby solving it via flow-based methods.

ABSTRACT

We apply the parabolic flow method to solving complex quotient equations on closed Kähler manifolds. We study the parabolic equation and prove the convergence. As a result, we solve the complex quotient equations.

Motivation & Objective

• To solve complex quotient equations of the form $\chi_u^k \wedge \omega^{n-k} = \psi \chi_u^l \wedge \omega^{n-l}$ on closed Kähler manifolds using parabolic flow methods.
• To establish the existence of a long-time solution to the parabolic flow $\partial_t u = \log\left(\frac{\chi_u^k \wedge \omega^{n-k}}{\chi_u^l \wedge \omega^{n-l}}\right) - \log\psi$ with initial condition $u(x,0) = 0$.
• To prove that the normalized solution $\hat{u}$ converges smoothly to a limit function $\hat{u}_\infty$ solving a modified equation with a constant $b$.
• To extend the parabolic method to complex quotient equations beyond the Monge-Ampère and Donaldson cases, generalizing previous results via flow techniques.

Proposed method

• Utilizes a parabolic flow equation $\partial_t u = \log\left(\frac{\chi_u^k \wedge \omega^{n-k}}{\chi_u^l \wedge \omega^{n-l}}\right) - \log\psi$ to evolve the potential $u$ over time.
• Applies the Alexandroff-Bakelman-Pucci (ABP) maximum principle locally in time to control $L^\infty$ estimates and prevent blow-up as $t \to \infty$.
• Employs $C^2$ $\mathcal{C}$-subsolution $\underline{u}$ to ensure uniform lower bounds and control the behavior of the flow.
• Uses second-order estimates based on the method of Hou, Ma, and Wu, with improvements to a key lemma from [25] to handle the fully nonlinear structure.
• Applies gradient estimates via blow-up arguments inspired by Dinew and Kolodziej and Gill, ensuring uniform control on $|\nabla u|$.
• Applies Evans-Krylov theorem and Schauder estimates to derive $C^\infty$ bounds on $[0,\infty)$, enabling long-time existence and convergence.

Experimental results

Research questions

• RQ1 Can the parabolic flow method be successfully applied to solve complex quotient equations beyond the Monge-Ampère and Donaldson cases?
• RQ2 Under what conditions does the parabolic flow $\partial_t u = \log\left(\frac{\chi_u^k \wedge \omega^{n-k}}{\chi_u^l \wedge \omega^{n-l}}\right) - \log\psi$ converge smoothly to a solution?
• RQ3 How can the time-dependent nature of the ABP estimate be overcome to ensure uniform $L^\infty$ control and long-time existence?
• RQ4 What role does the $\mathcal{C}$-subsolution play in ensuring the solvability of the complex quotient equation via parabolic flow?
• RQ5 Is the convergence of the normalized flow $\hat{u}$ to a smooth limit function $\hat{u}_\infty$ guaranteed under the given conditions?

Key findings

• The parabolic flow $\partial_t u = \log\left(\frac{\chi_u^k \wedge \omega^{n-k}}{\chi_u^l \wedge \omega^{n-l}}\right) - \log\psi$ admits a long-time solution on $[0,\infty)$ under the existence of a $\mathcal{C}$-subsolution and $\psi \geq c$.
• The normalized solution $\hat{u}$ converges smoothly in $C^\infty$ to a limit function $\hat{u}_\infty$ as $t \to \infty$.
• The limit $\hat{u}_\infty$ solves the equation $\frac{\chi^{k}_{\hat{u}_\infty} \wedge \omega^{n-k}}{\chi^{l}_{\hat{u}_\infty} \wedge \omega^{n-l}} = e^b \psi$ for some unique real number $b$.
• The time derivative of $\hat{u}$ decays exponentially: $\left|\frac{\partial \hat{u}}{\partial t}\right| \leq C e^{-c_0 t}$ for some $c_0 > 0$, ensuring convergence.
• The $L^\infty$ estimate is preserved via a local-in-time application of the ABP principle, avoiding blow-up issues from time-dependent bounds.
• The convergence result is established via standard arguments following Cao and Gill, relying on exponential decay of the time derivative and normalization to zero mean.