Developments of geometry and analysis for measurable Riemannian structures on fractals

2014 Fiscal Year Final Research Report

• Project/Area Number: 25887038
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Kobe University
• Principal Investigator: KAJINO Naotaka 神戸大学, 理学(系)研究科(研究院), 助教 (90700352)
• Project Period (FY): 2013-08-30 – 2015-03-31
• Keywords: フラクタル解析 / ディリクレ形式 / ラプラシアン / 固有値漸近挙動 / コンヌのトレース定理 / 測度論的リーマン構造 / アポロニウスの詰め込み

Outline of Final Research Achievements

In this research the author has studied asymptotic behavior of the distributions of the eigenvalues of Laplacians (the eigenfrequencies) on fractals and has proved the following assertions:
Connes' trace theorem, which characterizes the notion of volume as an operator-theoretic paraphrase of eigenvalue asymptotics, holds for a large class of Laplacians on fractals including the case of the measurable Riemannian structure on the Sierpinski gasket, and surface area can be also characterized in a similar manner.
Moreover, for the measurable Riemannian structure on a classical fractal called the Apollonian gasket, the author has proved several fundamental facts such as the self-adjointness of the associated Laplacian, the discreteness of the eigenvalue distribution of the Laplacian, and estimates of the smallest eigenvalue.

Free Research Field

フラクタル及び測度距離空間上の解析学