次椭圆边值问题与$G$ -Fredholm性质
Subelliptic boundary value problems and The $G$-Fredholm Property
摘要：
设$G$是幺模李群，$X$是带边界的紧致流形，$M$是主丛$G\to M\to X$的全空间，使得$M$也是满足局部次椭圆估计的复流形.在这项工作中，我们表明，如果$G$行为的全纯变换在$M$，那么拉普拉斯$\square=\bar\partial^{*}\bar\partial+\bar\partial\bar\partial^{*}$的$M$有以下性质：核的$\square$限制的形式$Λ^{p,q}$与$q>0$是一个封闭的，$G$不变的子空间在$L^{2}(M,Λ^{p,q})$有限$G$维。其次，我们表明，如果$q>0$，那么图像的$\square$包含一个封闭的，$G$不变的子空间有限余维$L^{2}(M,Λ^{p,q})$。这两个性质合在一起等于说$\square$是$G$ - Fredholm算子。在类似的情况下，边界拉普拉斯算子$\square_b$具有类似的性质。
英文摘要：
Let $G$ be a unimodular Lie group, $X$ a compact manifold with boundary, and $M$ be the total space of a principal bundle $G\to M\to X$ so that $M$ is also a complex manifold satisfying a local subelliptic estimate. In this work, we show that if $G$ acts by holomorphic transformations in $M$, then the Laplacian $\square=\bar\partial^{*}\bar\partial+\bar\partial\bar\partial^{*}$ on $M$ has the following properties: The kernel of $\square$ restricted to the forms $Λ^{p,q}$ with $q>0$ is a closed, $G$-invariant subspace in $L^{2}(M,Λ^{p,q})$ of finite $G$-dimension. Secondly, we show that if $q>0$, then the image of $\square$ contains a closed, $G$-invariant subspace of finite codimension in $L^{2}(M,Λ^{p,q})$. These two properties taken together amount to saying that $\square$ is a $G$-Fredholm operator. In similar circumstances, the boundary Laplacian $\square_b$ has similar properties.
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