Spectral convergence under bounded Ricci curvature

2015/10/19 by Shouhei Honda, Honda, Shouhei · 4 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research

paper · pdf · doi:10.48550/arxiv.1510.05349

Abstract

For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are generalizations of Cheeger-Colding's spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. As a corollary we show the upper semicontinuity of the first Betti numbers with respect to the Gromov-Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. We also define the Riemannian curvature tensor, the Ricci curvature, and the scalar curvature of the limit space, and show their properties. In particular the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli's one and Lott's Ricci measure. Moreover a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott's question on the behavior of the scalar curvature with respect to the Gromov-Hausdorff topology.

Citations

• Regularity of Einstein manifolds and the codimension 4 conjecture
• Ricci curvature for metric-measure spaces via optimal transport
• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• Differential equations on riemannian manifolds and their geometric applications
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• Collapsing of Riemannian manifolds and eigenvalues of Laplace operator
• Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

Cited by

• The harmonic 2-forms on K3 surfaces converging to a flat 4-dimensional orbifold
• Mosco-convergence of Cheeger energies on varying spaces satisfying curvature dimension conditions
• Stability of local Riemannian Ricci curvature lower bounds
• Strongly continuous fields of operators over varying Hilbert spaces

Related

• Ricci curvature and L^p-convergence
• Intrinsic flat and Gromov-Hausdorff convergence of manifolds with Ricci\n curvature bounded below
• Spectral convergence of graph Laplacians with Ricci curvature bounds and in non-collapsed Ricci limit spaces
• Ricci curvature and convergence of Lipschitz functions
• Intrinsic flat convergence with bounded Ricci curvature