2020 Fiscal Year Annual Research Report
Geometric analysis and comparison geometry on weighted manifolds
| Project/Area Number |
20J11328
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| Research Institution | Osaka University |
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Principal Investigator |
Mai Cong Hung 大阪大学, 理学研究科, 特別研究員(PD)
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| Project Period (FY) |
2020-04-24 – 2022-03-31
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| Keywords | Bakry-Ledoux inequality / isoperimetric inequality / needle decomposition / weighted manifolds / noncompact manifolds / weighted Ricci curvature / L1-estimate |
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| Outline of Annual Research Achievements |
We have completed the research revision on the quantitative estimate of Bakry-Ledoux inequality during this research period. We gave an upper bound of the volume of the symmetric difference between a Borel set and a sub-level set of the guiding function arising in the needle decomposition associated with the isoperimetric minimizer in terms of the deficit in Bakry-Ledoux's Gaussian isoperimetric inequality. We also published an extension work of this research to provide several further applications of our detailed estimates. We give an L1-estimate to show that the push-forward of the reference measure by the guiding function is close to the Gaussian measure. This work is a collaboration with Shin-ichi Ohta (Osaka University).
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The work on the quantitative estimate of Bakry-Ledoux inequality is as expected, but the extension to the setting of manifolds of negative effective dimension is still in progress.
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| Strategy for Future Research Activity |
I will continue the study of quantitative estimates of isoperimetric inequalities in noncompact manifolds of negative effective dimension. Furthermore, I will investigate the almost splitting phenomenon on CD(K, ∞)-weighted manifolds. Our previous results open a question to study the convergence theory of noncompact manifolds under the weighted Ricci curvature bound. We plan to use the quantitative estimate of Bakry-Ledoux inequality to verify the following conjecture: If a sequence of CD(K, ∞)-weighted manifolds have the isoperimetric deficit tending to zero, then this sequence would converge to a product space of the 1-dimensional Gaussian space in the Gromov-Hausdorff topology (or a weaker distance such as the concentration topology).
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