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Weak Solutions to the complex Monge-Ampère flows on compact Kähler manifolds : general measures on the right-hand side

Bowoo Kang

Abstract

We show the existence of a bounded solution to the Cauchy problem for the complex Monge-Ampère flow on a compact Kähler manifold, with the right-hand side of the form $dt \wedge dμ$ where $dμ$ is dominated by a Monge-Ampère measure of a Hölder continuous quasi-plurisubharmonic function. We also prove that for a given semi-positive big from $θ$, the $t$-slice of the solution is locally Hölder continuous on $\rm{Amp(θ)}$ for all $t \in (0, T)$. Next, we prove a comparison principle when $dμ$ is dominated by a Monge-Ampère measure of a bounded quasi-plurisubharmonic function, which implies the uniqueness of the solution.

Weak Solutions to the complex Monge-Ampère flows on compact Kähler manifolds : general measures on the right-hand side

Abstract

We show the existence of a bounded solution to the Cauchy problem for the complex Monge-Ampère flow on a compact Kähler manifold, with the right-hand side of the form where is dominated by a Monge-Ampère measure of a Hölder continuous quasi-plurisubharmonic function. We also prove that for a given semi-positive big from , the -slice of the solution is locally Hölder continuous on for all . Next, we prove a comparison principle when is dominated by a Monge-Ampère measure of a bounded quasi-plurisubharmonic function, which implies the uniqueness of the solution.
Paper Structure (14 sections, 26 theorems, 214 equations)

This paper contains 14 sections, 26 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parabolic Monge-Ampère operator
  4. Time derivatives of parabolic potentials
  5. Estimates on the solutions
  6. Parabolic comparison principle
  7. Convergence Lemmas
  8. Existence Result
  9. Bounded solution
  10. Hölder continuous solution
  11. Uniqueness Result
  12. Mixed type inequalities
  13. Boundary behavior of supersolutons
  14. General comparison principle

Key Result

Theorem 1.2

Let $u_0 \in PSH(X, \omega_0) \cap L^{\infty}(X)$. Assume that there exist a Hölder continuous $\omega_X$-psh function $\phi$ and a constant $C > 0$ satisfying Then, there exists $u \in \mathcal{P}(X_T, \omega_t) \cap L^{\infty}(X_T)$ satisfying where the Hölder exponent of $u(t, \cdot)$ does not depend on $t$. Moreover, $u$ is jointly continuous on $(0, T) \times {\rm{Amp}}(\theta)$.

Theorems & Definitions (56)

  • Definition 1.1: GLZ20
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: GLZ20
  • Definition 2.2: GLZ20
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 46 more