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Topological stability of RCD spaces

Research Project

Project/Area Number 23KJ0204
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeMulti-year Fund
Section国内
Review Section Basic Section 11020:Geometry-related
Research InstitutionTohoku University

Principal Investigator

PENG YUANLIN  東北大学, 理学研究科, 特別研究員(DC1)

Project Period (FY) 2023-04-25 – 2026-03-31
Project Status Granted (Fiscal Year 2023)
Budget Amount *help
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2025: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2024: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2023: ¥800,000 (Direct Cost: ¥800,000)
Keywordsmetric geometry / RCD spaces / Green function / rigidity
Outline of Research at the Start

Our research is intended to prove a topological stability theorem for a special class of metric measure spaces, namely $RCD(K,N)$ spaces which, roughly speaking, are metric measure spaces with Ricci curvature bounded below by $K$ and dimension bounded above by $N$ in some synthetic sense. Our desired result is that if an $RCD(K,N)$ space is similar to a Riemannian manifold (in mGH-sense), then we can deform the $RCD(K,N)$ space into the manifold with arbitrarily small deformation.

Outline of Annual Research Achievements

In the past academic year, the author focused on the analysis and geometry of RCD spaces.
As planned at the beginning of this project, the author has read the literature mentioned in the research plan. Besides, he also found some other recent results that might be useful. He believes there is some way to solve the problem hidden behind these results.
As for research acievements, the author and his supervisor, Prof. Honda, established some properties of Green functions on metric measure spaces with non-negative Ricci curvature, including a sharp gradient estimate, a rigidity theorem to cone, and the quantitative version of rigidity theorem. These results are helpful for further study of elliptic PDEs on RCD spaces.

Current Status of Research Progress
Current Status of Research Progress

1: Research has progressed more than it was originally planned.

Reason

In the plan proposed at the beginning of the project, the main purpose of the first year is to learn related work that might be helpful to prove the desired Laplacian comparison conjecture. The author has already read the literature listed in the research plan. And moreover, he also found some other directions that may work, for instance, some results defining the mean curvature of the boundary of subsets in RCD spaces using the isoperimetric property.
Besides, the author is also investigating along the direction of analysis and PDE. That is why he also focused on study of Green functions on RCD spaces. These published results can be regarded as byproducts of this project.

Strategy for Future Research Activity

As there are hopeful methods already come into the author's view, it is time to attack the main question, namely the Laplacian comparison conjecture. As observed by the author and his collaborators, the main difficulty is the analysis near the boundary. There are two possible directions. The first one is to get this result via methods of analysis and PDE. The key point of this way is to extend the space compactiblly with the boundary. This needs further regularity result of the boundary. The second one is to work more on the geometric aspect. And this way requires better definition and properties of the tensors, for instance, second fundamental form and mean cuvature, on the boundary. From the author's view, regularity of boundary is also needed here.

Report

(1 results)
  • 2023 Research-status Report
  • Research Products

    (2 results)

All 2024 2023

All Journal Article (1 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 1 results) Presentation (1 results) (of which Int'l Joint Research: 1 results)

  • [Journal Article] Sharp gradient estimate, rigidity and almost rigidity of Green functions on non-parabolic RCD (0, N) spaces2024

    • Author(s)
      Shouhei Honda, Yuanlin Peng
    • Journal Title

      Proceedings of the Royal Society of Edinburgh Section A: Mathematics

      Volume: First view Pages: 1-54

    • DOI

      10.1017/prm.2024.131

    • Related Report
      2023 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Almost rigidity of Green functions on RCD(0,N) spaces2023

    • Author(s)
      Yuanlin Peng
    • Organizer
      Geometry & Probability 2023
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research

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Published: 2023-04-26   Modified: 2024-12-25  

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