Ricci curvature and $L^p$-convergence

By Shouhei Honda. 2012

63 citations recorded by the source corpus.

We give the definition of $L^p$-convergence of tensor fields with respect to the Gromov-Hausdorff topology and several fundamental properties of the convergence. We apply this to establish a Bochner-type inequality which keeps the term of Hessian on the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a lower Ricci curvature bound and to give a geometric explicit formula for the Dirichlet Laplacian on a limit space defined by Cheeger-Colding. We also prove a continuity of the first eigenvalues of the p-Laplacian with respect to the Gromov-Hausdorff topology.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c291048c3ed945dfe6f76709b70b68daad5ae9b5

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

New stability results for sequences of metric measure spaces with uniform Ricci bounds from below (2016, background): “p |ϕ| ◦fi, from the strong L2-convergence of p ϕ± ◦ fi to p ϕ± ◦ f and the additivity of L2-strong convergence (proved in (b)) we get the result. 7 (b) The case p>1 is dealt with, for instance, in [H15], see Corollary 3.26 and Proposition 3.31 therein. In order to prove additivity for p = 1 we ca”
Local spectral convergence in RCD∗(K,N) spaces (2017, background): “d the following question: what happens in the result (⋆) when the sequence of spaces is collapsed? We can give a positive answer to the question in the RCD∗-setting; (Xi,di,xi,mi) mGH→ (X,d,x,m) (see [H15] for the corresponding results in the Ricci limit setting). Then, combining this fact with the ”