No abstract

We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative Bakry-Émery Ricci curvature. We derive some universal inequalities among eigenvalues of the weighted Laplacian on such manifolds. These inequalities are quantitative versions of the previous theorem by the author with Shioya. We also study some geometric quantity, called multi-way isoperimetric constants, on such manifolds and obtain similar universal inequalities among them. Multi-way isoperimetric constants are generalizations of the Cheeger constant. Extending and following the heat semigroup argument by Ledoux and E. Milman, we extend the Buser-Ledoux result to the k-th eigenvalue and the k-way isoperimetric constant. As a consequence the k-th eigenvalue of the weighted Laplacian and the k-way isoperimetric constant are equivalent up to polynomials of k on closed weighted manifolds of nonnegative Bakry-Émery Ricci curvature.

Abstract. Chung-Grigor'yan-Yau's inequality describes upper bounds of eigenvalues of Laplacian in terms of subsets ("input") and their volumes. In this paper we will show that we can reduce ginputh in Chung-Grigor'yan-Yau's inequality in the setting of Alexandrov spaces satisfying CD(0, ∞). We will also discuss a related conjecture for some universal inequality among eigenvalues of Laplacian.

The property of measure concentration is that an arbitrary 1-Lipschitz function f : X → R on an mm-space X is almost close to a constant function. In this paper, we prove that if such a concentration phenomenon arise, then any 1-Lipschitz map f from X to a space Y with a doubling measure also concentrates to a constant map. As a corollary, we get any 1-Lipschitz map to a Riemannian manifold with a lower Ricci curvature bounds also concentrates to a constant map.

In this paper we prove the following. Let Σ be an n-dimensional closed hyperbolic manifold and let g be a Riemannian metric on Σ × S 1 . Given an upper bound on the volumes of unit balls in the Riemannian universal cover ( Σ × S 1 , g), we get a lower bound on the area of the Z 2 -homology class [Σ × * ] on Σ × S 1 , proportional to the hyperbolic area of Σ. The theorem is based on a theorem of Guth and is analogous to a theorem of Kronheimer and Mrowka involving scalar curvature.