論文の公開元へRigidity for the isoperimetric inequality of negative effective dimension on weighted Riemannian manifolds
概要
Isoperimetric inequality comparing the volume of a set and the perimeter of its boundary is a classical and important topic in geometry and analysis. On weighted Riemannian manifolds (Riemannian manifolds equipped with arbitrary measures), the rigidity problem of the isoperimetric inequality is of interest because the appearance of densities on their structures arises new perspective compared with the classical setting of Euclidean spaces and (unweighted) Riemannian manifolds.
On a weighted Riemannian manifold, the Ricci curvature is modified into the weighted Ricci curvature RicN involving a real parameter N which is called the effective dimension. In the modified curvature bound RicN ≥ K, the parameters K and N could be regarded as “a lower bound of the Ricci curvature” and “an upper bound of the dimension”, respectively. Geometric analysis for the weighted Ricci curvature bound including the isoperimetric inequality has been intensively studied by Bakry and his collaborators via the Γ-calculus in the case where N is larger than the dimension of the manifold. Recently it turned out that there is a rich theory also for negative N, though this range may seem strange due to the above interpretation of N as an upper dimension bound. In his previous paper, Mai studied the rigidity of the spectral gap (Poincare inequality) under the condition RicN ≥ K for K > 0 and N < −1, and showed that the sharp spectral gap is achieved only if the space is isometric to a certain warped product of hyperbolic nature. In this thesis, Mai continued this study to the rigidity problem of the isoperimetric inequality under the same curvature bound.
The isoperimetric inequality on weighted Riemannian manifolds satisfying RicN ≥ K was intensively studied by E. Milman via (classical but technical) geometric measure theory. The isoperimetric inequality could also be verified in a gentle way called the needle decomposition (or localization), developed on Riemannian manifolds by Klartag. The idea is to foliate a high dimensional manifold into geodesics (needles) while preserving the curvature bound condition. Firstly, on needles, we obtain the one-dimensional version of the rigidity problem. By taking integration all over the needles, the main theorem asserts that the weighted manifold is necessarily isometric to the warped product of hyperbolic nature.
This result is directly related to the lower bound problem of the first nonzero eigenvalue in the previous paper, because the hyperbolic sine of the Lipschitz function used to construct the needle decomposition (called the guiding function) turns out to be an eigenfunction associated with the first nonzero eigenvalue of the Laplacian achieving the sharp spectral gap.
