Geometry of Laplace–Beltrami Operator on a Complete Riemannian Manifold
Geometry of Laplace–Beltrami Operator on a Complete Riemannian Manifold
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完全黎曼流形上拉普拉斯-贝尔特拉米算子的几何
DOI：
10.2969/aspm/02210347
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发表时间：
1993
期刊：
影响因子：
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通讯作者：
H. Urakawa
中科院分区：
文献类型：
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作者：
H. Urakawa
这是一篇综述性论文，介绍了完备黎曼流形上拉普拉斯-贝尔特拉米算子的解析和几何方面的最新进展。从黎曼几何观点出发的系统处理方法已经在Berger，Gauduchon & Mazet [1971]，Kotake，Maeda，Ozawa & Urakawa [1981]，Berard & Berger [1983]，Berard [1986]，Chavel [1984]，Gilkey [1984]和Sunada [1988]中得到了体现。但他们主要关注的是紧凑的情况下，除了Chavel '84]。本文主要讨论非紧完备黎曼流形谱几何的最新发展。这些材料似乎可以分为三部分：（1）拉普拉斯算子的（本质）谱的分布，（2）完备黎曼流形的热核，（3）这种流形上的调和函数和绿色函数。更确切地说，（1）在§3中，我们主要讨论非紧完备黎曼流形的拉普拉斯算子的（本质）谱底的估计。(2)在§4中，继Ito [1988]，Dodziuk [1983]之后，我们构造了非紧完备黎曼流形的（极小）热核，并在一定的曲率条件下给出了这类热核的唯一性和估计. (3)在§5中，我们将讨论完备流形上的正调和函数、Martin边界和调和函数的Liouville型定理。在此，我们向主编K教授表示衷心的感谢。盐滨教授给了我们发表这篇文章的机会
This is a survey paper on recent developments of analytic and geometric aspects of the Laplace-Beltrami operator on a complete Riemannian manifold. Systematic treatments from a Riemannian geometric viewpoint have been already appreared in Berger, Gauduchon & Mazet ['71], Kotake, Maeda, Ozawa & Urakawa ['81], Berard & Berger ['83], Berard ['86], Chavel ['84], Gilkey ['84] and Sunada ['88]. But they are mainly concerned with compact case, except Chavel ['84]. In this paper, we shall focus on recent developments of spectral geometry of a noncompact complete Riemannian manifold. It seems that the materials may be divided into three parts: (1) the distribution of the (essential) spectrum of the Laplacian, (2) the heat kernel of a complete Riemannian manifold, and (3) harmonic functions, and Green functions on such a manifold. More precisely, (1) in §3, we treat mainly results on estimates of the bottom of the (essential) spectrum of the Laplacian of a noncompact complete Riemannian manifold. (2) In §4, following Ito ['88], Dodziuk ['83], we construct the (minimal) heat kernel of a noncompact complete Riemannian manifold, and show results on uniqueness and estimates of such heat kernel, under certain curvature conditions. (3) In §5, we will treat positive harmonic functions, the Martin boundary, and Liouville type theorems for harmonic functions on complete manifolds. We express our sincere gratitude to the editor, Professor K. Shiohama who gave us an opportunity of publishing this note, Professor