复Monge-Ampere方程的几何稳定性

中文摘要 AI 辅助

设$X$为紧致Kähler流形。Kolodziej关于复Monge-Ampere方程的解析稳定性定理指出：对于同一上同调类中的任意Kähler度量$\omega$和$\omega'$，若其体积测度在$L^p(X)$（对某个$p>1$）中有界且在$L^1(X)$中接近，则其Kähler势在$L^\infty(X)$中接近。本文建立了复Monge-Ampère方程的几何稳定性，即体积测度的$L^1$-接近性蕴含由$\omega$和$\omega'$诱导的距离函数的$L^\infty$-接近性。因此，我们证明了任何体积测度在$L^p$（对某个$p>1$）中有界且Ricci电流下有界的非光滑Kähler电流诱导出唯一的度量空间，该空间为紧致RCD空间，且与$X$本身同胚。

英文摘要

Let $X$ be a compact Kahler manifold. The analytic stability theorem of Kolodziej for complex Monge-Ampere equation states that for any Kahler metrics $ω$ and $ω'$ in the same cohomology class, if their volume measures are bounded in $L^p(X)$ (for some $p>1$) and close in $L^1(X)$, then their Kahler potentials are close in $L^\infty(X)$. In this paper, we establish the geometric stability for complex Monge-Ampère equations that $L^1$-closeness of volume measures implies $L^\infty$-closeness for the induced distance functions by $ω$ and $ω'$. Consequently, we prove that any non-smooth Kahler current with volume measure bounded in $L^p$ (for some $p>1$) and Ricci current bounded below induces a unique metric space, which turns out to be a compact RCD space homeomorphic to $X$ itself.

发表机构

• Rutgers University（罗格斯大学）

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