Compactification of Riemannian manifolds and embeddings of graphs

2015 Fiscal Year Final Research Report

• Project/Area Number: 24540072
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kanazawa University
• Principal Investigator: Kasue Atsushi 金沢大学, 数物科学系, 教授 (40152657)
• Co-Investigator(Kenkyū-buntansha): Hattori Tae 石川工業高等専門学校, 一般教育, 講師 (40569365)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: リーマン多様体 / ネットワーク / 理想境界 / ディリクレ形式 / ディリクレエネルギー有限写像 / ランダムウォーク / スペクトルギャップ / 双曲埋め込み

Outline of Final Research Achievements

We study a connected nonparabolic, or transient network compactified with the Kuramochi boundary, and show that the random walk converges almost surely to a random variable valued in the harmonic boundary, and a function of finite Dirichlet energy converges along the random walk to a random variable almost surely　and in L2. We also give integral representations of solutions of Poisson equations on the Kuramochi compactification. 　We also study finite connected graphｓ which admit quasi monomorphismｓ to hyperbolic spaces and give geometric bounds for the Cheeger constants in terms of the volume, an upper bound of the degree, and the quasi monomorphism. Moreover we develop a potential theory of nonlinear networks in the frame work of modular sequence spaces.

Free Research Field

数物科学系