论文标题
能量功能的第二个与规范偏光流的家族相关的变化
Second Variation of Energy Functions associated to Families of Canonically Polarized Manifolds
论文作者
论文摘要
令$π：\ Mathcal {x} \ t to s $为复杂的歧管$ s $上的典型两极化流形的全态家族，而$ f：\ mathcal {x} \ to n $ to n $ to n $平滑地图进入riemannian cristianian cormord $ n $。考虑能量函数$ e：s \ to \ mathbb {r} $，将$ z \ in s $分配给地图的dirichlet $ f | _ {\ mathcal {x} {x} _z}：\ mathcal {x}} _z} _z \ to n $ $ z $。在本文中，我们计算了能量函数$ e $的第二个变体公式。结果，我们表明，如果$ n $具有非阳性的综合截面曲率，并且每张映射$ f | _ {\ Mathcal {x} _z}：\ Mathcal {x} _z \ to n $是谐的，那么$ e：S \ to \ s \ to \ mathbb {r} $ is plurisubhmarmonic。
Let $π:\mathcal{X}\to S$ be a holomorphic family of canonically polarized manifolds over a complex manifold $S$, and $f:\mathcal{X}\to N$ a smooth map into a Riemannian manifold $N$. Consider the energy function $E: S\to \mathbb{R}$ that assigns $z\in S$ to the Dirichlet energy of the map $f|_{\mathcal{X}_z}:\mathcal{X}_z\to N$, where $\mathcal{X}_z=π^{-1}(z)$ is the fiber over $z$. In this article, we compute the second variation formula for the energy function $E$. As a result, we show that if $N$ is of non-positive complexified sectional curvature and every map $f|_{\mathcal{X}_z}:\mathcal{X}_z\to N$ is harmonic, then $E:S\to \mathbb{R}$ is plurisubharmonic.
