CR eigenvalue estimate and Kohn-Rossi cohomology.
CR eigenvalue estimate and Kohn-Rossi cohomology.
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DOI：
10.4310/jdg/1717348874
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发表时间：
2019-05
影响因子：
2.5
通讯作者：
Zhiwei Wang;Xiangyu Zhou
中科院分区：
文献类型：
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作者：
Zhiwei Wang;Xiangyu Zhou
设X是一个紧连通的CR流形，其横截CR S ^1-作用的维数为2n-1，且X是弱伪凸的.设$\Box_B$为$\overline{\partial}_B$-拉普拉斯算子。$\Box_B$的特征值估计是CR几何和分析中的一个基本问题。本文得到了X上光滑$（n-1，q）$-形式的$m$阶Fourier分量上$\Box_B$的小于或等于$\lambda$的特征值个数的精确估计，其中$m\in \mathbb{Z}_+$且$q= 0，1，\cdots，n-1$.这里的锐意味着相对于$m$的增长顺序是锐的。特别地，当$\lambda=0$时，我们得到了$H^{n-1，q}_B（X）$的$m$阶Fourier分量$H ^{n-1，q}_{B，m}（X）$的增长性的渐近估计为$m \rightarrow +\infty$.此外，我们还建立了Kohn-Rossi上同调Fourier分量的Serre型对偶定理。作为副产品，建立了$ m\in \mathbb{Z}_+$的傅立叶分量$H^{0，q}_{B，-m}（X）$的维数的渐近增长。与以前的结果相比，在这个领域中，估计$\lambda=0$已经提高了很大的相应的估计Hsiao和李。给出了我们主要结果的应用，包括莫尔斯型不等式，渐近Riemann-Roch型定理，Grauert-Riemenscheider型判别准则，以及我们主要结果的一个轨道形式，它回答了一个公开问题.
Let $X$ be a compact connected CR manifold with a transversal CR $S^1$-action of dimension $2n-1$, which is only assumed to be weakly pseudoconvex. Let $\Box_b$ be the $\overline{\partial}_b$-Laplacian. Eigenvalue estimate of $\Box_b$ is a fundamental issue both in CR geometry and analysis. In this paper, we are able to obtain a sharp estimate of the number of eigenvalues smaller than or equal to $\lambda$ of $\Box_b$ acting on the $m$-th Fourier components of smooth $(n-1,q)$-forms on $X$, where $m\in \mathbb{Z}_+$ and $q=0,1,\cdots, n-1$. Here the sharp means the growth order with respect to $m$ is sharp. In particular, when $\lambda=0$, we obtain the asymptotic estimate of the growth for $m$-th Fourier components $H^{n-1,q}_{b,m}(X)$ of $H^{n-1,q}_b(X)$ as $m \rightarrow +\infty$. Furthermore, we establish a Serre type duality theorem for Fourier components of Kohn-Rossi cohomology which is of independent interest. As a byproduct, the asymptotic growth of the dimensions of the Fourier components $H^{0,q}_{b,-m}(X)$ for $ m\in \mathbb{Z}_+$ is established. Compared with previous results in this field, the estimate for $\lambda=0$ already improves very much the corresponding estimate of Hsiao and Li . We also give appilcations of our main results, including Morse type inequalities, asymptotic Riemann-Roch type theorem, Grauert-Riemenscheider type criterion, and an orbifold version of our main results which answers an open problem.