DISCRETIZATION OF VECTOR BUNDLES AND ROUGH LAPLACIAN

摘要

Let ,M.(rn, K, ro) be the set of all compact connected m-dimensional manifolds (M, g) such that Ricci(M.g) > - (m - I) Kg and Inj(M,g) > TQ > 0, Let ?(n,ki,k2) be the set of all Riemannian vector bundles (E,'V) of real rank n with RE < k and d*RE < fc2 For any vector bundle E E ?{n, k,k2) with harmonic curvature or with complex rank one, over any M e M-(m, K, ro) and for any discretization X of M of mesh 0 < e < ^ro, we construct a canonical twisted Laplacian A A and a potential V depending only on the local geometry of E and M such that we can compare uniformly the spectrum of the rough Laplacian A associated to the connection of E and the spectrum of AA + V. We show that there exist constants c, cl > 0 depending only on the parameters of M(rn, re, r0) and ?(n, fci, ?2) such that c'k(X, A, V) < k(E) < ck(X, A, V), where Afc( ) denotes the kth eigenvalue of the considered operators (fe < njX|). For flat vector bundles, we show that the potential is zero, A^ turns out to be a discrete magnetic Laplacian and we relate Ai (E) to the holonomy of E.