论文标题
最佳$ l^2 $开放类型扩展
Optimal $L^2$ Extensions of Openness Type
论文作者
论文摘要
我们研究以下最佳$ l^2 $扩展问题类型：给定复杂的歧管$ m $，一个封闭的子变量$ s \ subset m $和一个holomorphic vector bundle $ e \ rightarrow m $，对于任何$ l^2 $ holomorphic $ holomorphic $ f $在某些$ $ $ $ u $ $ u $ u $ l^2 $ l^2 $ f $上定义的$ l^2 $ lomor， $ f | _s = f | _s $，$ m $上的$ l^2 $ norm of $ f $的$ f $由$ u $上的$ l^2 $ norm of $ f $。 在回答上述问题时，我们证明了在弱伪vavexkähler歧管上开放类型的最佳$ l^2 $扩展定理，该歧管在此问题上概括了几个已知结果。此外，我们证明了某些最小$ l^2 $扩展名的产品属性，并为上述$ l^2 $扩展定理的版本提供了替代证明。我们还向通常的最佳$ l^2 $扩展问题和Suta猜想的平等部分提供了一些应用程序。
We study the following optimal $L^2$ extension problem of openness type: given a complex manifold $M$, a closed subvariety $S\subset M$ and a holomorphic vector bundle $E\rightarrow M$, for any $L^2$ holomorphic section $f$ defined on some open neighborhood $U$ of $S$, find an $L^2$ holomorphic section $F$ on $M$ such that $F|_S = f|_S$, and the $L^2$ norm of $F$ on $M$ is optimally controlled by the $L^2$ norm of $f$ on $U$. Answering the above problem, we prove an optimal $L^2$ extension theorem of openness type on weakly pseudoconvex Kähler manifolds, which generalizes a couple of known results on such a problem. Moreover, we prove a product property for certain minimal $L^2$ extensions and give an alternative proof to a version of the above $L^2$ extension theorem. We also present some applications to the usual optimal $L^2$ extension problem and the equality part of Suita's conjecture.
