论文标题
FiberwiseKähler-Ricci流向有限的强烈伪convex域的家族
Fiberwise Kähler-Ricci flows on families of bounded strongly pseudoconvex domains
论文作者
论文摘要
令$π：\ mathbb {c}^n \ times \ times \ mathbb {c} \ rightarrow \ rightArrow \ mathbb {c} $是第二个因素的投影映射，让$ d $是$ \ mathbb {c}^{n+1} $ y Mathbb {c}^{n+1} $ y y y Mathbb { $ d_y：= d \capπ^{ - 1}（y）$是$ \ mathbb {c}^n $中的平滑界限强pseudoconvex域，并且彼此之间是差异的。根据Chau的定理，Kähler-ricci流具有长时间的解决方案$ω_y（t）$ tober $ d_y $。这个流量的家族在总空间$ d $上引起了平滑的真实（1,1）-form $ω（t）$，其限制对光纤$ d_y $满足$ω（t）\ vert_ {d_y} =ω_y（t）$。在本文中，我们证明$ω（t）$对于$ω（0）$是正面的所有$ t> 0 $ in $ d $ in $ t> 0 $。作为推论，我们还证明，如果$ d $是$ \ mathbb {c}^{n+1} $，则$ d $的光纤kähler-inenstein公制是$ d $的正半明确。
Let $π:\mathbb{C}^n\times\mathbb{C}\rightarrow\mathbb{C}$ be the projection map onto the second factor and let $D$ be a domain in $\mathbb{C}^{n+1}$ such that for $y\inπ(D)$, every fiber $D_y:=D\capπ^{-1}(y)$ is a smoothly bounded strongly pseudoconvex domain in $\mathbb{C}^n$ and is diffeomorphic to each other. By Chau's theorem, the Kähler-Ricci flow has a long time solution $ω_y(t)$ on each fiber $D_y$. This family of flows induces a smooth real (1,1)-form $ω(t)$ on the total space $D$ whose restriction to the fiber $D_y$ satisfies $ω(t)\vert_{D_y}=ω_y(t)$. In this paper, we prove that $ω(t)$ is positive for all $t>0$ in $D$ if $ω(0)$ is positive. As a corollary, we also prove that the fiberwise Kähler-Einstein metric is positive semi-definite on $D$ if $D$ is pseudoconvex in $\mathbb{C}^{n+1}$.
