Study of metric measure spaces with curvature-dimension conditions and its applications to Riemannian geometry

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K13412
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kumamoto University
• Principal Investigator: Kitabeppu Yu 熊本大学, 大学院先端科学研究部(理), 准教授 (50728350)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 熱流

Outline of Final Research Achievements

It is known that we can use much techniques of calculus on RCD spaces, which can be applied to wide class including collapsed manifolds.
Roughly, we have the following two results: One is the Zhong-Yang type rigidity theorem for RCD spaces, and another is giving a definition of Ricci curvature on hypergraphs associated with the heat flow.

Free Research Field

測度距離空間の幾何

Academic Significance and Societal Importance of the Research Achievements

ハイパーグラフはネットワークのモデルなどとしてよく使われており、特に近年ハイパーグラフ上の熱の伝わりかたに関する研究もよくされている。ハイパーグラフ上に熱流、正確にいえば非線形ラプラシアンから定まるレゾルベント、を用いた曲率の定義を与えたことで、今後ハイパーグラフ上の現実的な問題に関しても応用があると期待される。