| Abstract: |
| The aim of this talk is to present the validity of weighted Ricci curvature whose dimensional parameter is equal to ``1 in view of extrinsic geometric analysis. It has been observed that the 1-weighted Ricci curvature exhibits singular behavior from the view point of the Cheeger-Gromoll splitting theorem and affine geometry. Recently, it has been also pointed out that its non-negativity is equivalent to the so-called substatic condition in the context of the Lorentzian geometry via conformal change of metric. After I review such developments, I will introduce several geometric consequences concerning stable weighted minimal hypersurfaces under a lower 1-weighted Ricci curvature bound. This talk is based on the joint work with Yasuaki Fujitani (University of Tokyo). |
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