Herbert Koch, Hart F. Smith and Daniel Tataru Sharp $L^p$ bounds on spectral clusters for Lipschitz metrics Amer. J. Math., 136(6): 1629-1663 2014 http://dx.doi.org/10.1353/ajm.2014.0039
Abstract: We establish Lp bounds on L2 normalized spectral clusters for self-adjoint elliptic Dirichlet forms with Lipschitz coefficients. In two dimensions we obtain best possible bounds for all 2 ≤ p ≤ ∞, up to logarithmic losses for 6 < p ≤ 8. In higher dimensions we obtain best possible bounds for a limited range of p.
Abstract: Let X be a finite graph. Let |V| be the number of its vertices and d be its degree. Denote by F1(X) its first spectral density function which counts the number of eigenvalues ≤λ2 of the associated Laplace operator. We provide an elementary proof for the estimate F1(X)(λ)−F1(X)(0)≤2⋅(|V|−1)⋅d⋅λ for 0≤λ<1 which has already been proved by Friedman (1996) [3] before. We explain how this gives evidence for conjectures about approximating Fuglede–Kadison determinants and L2-torsion.
Abstract: We prove generalized lower Ricci curvature bounds for warped products over complete Finsler manifolds. On the one hand our result covers a theorem of Bacher and Sturm concerning Euclidean and spherical cones (Bacher and Sturm [3]). On the other hand it can be seen in analogy to a result of Bishop and Alexander in the setting of Alexandrov spaces with curvature bounded from below (Alexander and Bishop, 2004 [2]). For the proof we combine techniques developed in these papers. Because the Finslerian warped product metric can degenerate we regard a warped product as metric measure space that is in general neither a Finsler manifold nor an Alexandrov space again but a space satisfying a curvature-dimension condition in the sense of Lott–Villani/Sturm.
Kazumasa Kuwada and Karl-Theodor Sturm Monotonicity of time-dependent transportation costs and coupling by reflection Potential Analysis, 39(3): 231-263 2013 http://dx.doi.org/10.1007/s11118-012-9327-4
Abstract: Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Émery’s curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distributions on spaceforms. As a corollary, we obtain a comparison theorem for the total variation between heat distributions. In addition, we show that our monotonicity is stable under the Gromov-Hausdorff convergence of the underlying space under a uniform curvature-dimension and diameter bound.