2023 Fiscal Year Research-status Report
Topological stability of RCD spaces
| Project/Area Number |
23KJ0204
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| Research Institution | Tohoku University |
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Principal Investigator |
PENG YUANLIN 東北大学, 理学研究科, 特別研究員(DC1)
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| Project Period (FY) |
2023-04-25 – 2026-03-31
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| Keywords | metric geometry / RCD spaces / Green function / rigidity |
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| 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.
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| 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.
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| 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.
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| Causes of Carryover |
I mainly used the grant to buy books this year, while the travelling expenses are not used. The reason is that my collaborators came to Japan to attend conferences, and I met with them there, thus travelling abroad was not necessary in the last year. However, it is still possible that I need to attend conferences abroad or meet with international collaborators. On the other hand, as Prof. Honda moved to Tokyo this year, I also need to go there to meet with him. Thus the grants are still needed.
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