Eigenvalue estimates and differential form Laplacians on Alexandrov spaces

Abstract

We give upper bounds on the eigenvalues of the differential form Laplacian on a compact Riemannian manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces. Under a local biLipschitz assumption on the Alexandrov space, which is conjecturally always satisfied, we show that the differential form Laplacian has a compact resolvent. We identify its kernel with an intersection homology group.

New features of the first eigenvalue on negatively curved spaces

“…The equality in (1.11) implies certain (radial) curvature rigidity and isometry between B r (x 0 ) and B κ r , see Cheng [8, Theorem 1.1]. The above-sketched consequences of Theorem 1.2 complement in several aspects the results concerning the first eigenvalue problem on compact Riemannian/Finsler manifolds developed by Ge and Shen [13], Lott [22], Shen, Yuan and Zhao [37], Wang and Xia [40], and Wu and Xin [41].…”

“…In the past half-century, McKean's and Cheng's results have become a continuing source of inspiration concerning the first eigenvalue problem on curved spaces; without seeking completeness, we recall the works of Carroll and Ratzkin [5], Chavel [6], Freitas, Mao and Salavessa [11], Gage [12], Hurtado, Markvorsen and Palmer [16], Li and Wang [20,21], Lott [22], Mao [24], Pinsky [31,32] and Yau [42], where various estimates and rigidity results concerning the equality in (1.5) are established.…”

Heat kernel bounds and Ricci curvature for Lipschitz manifolds

“…complete). Relevant works study differential forms [19,48,60], Hodge-de Rham's theorem and index theory [31,57], eigenvalue estimates [36], boundaryvalue problems [20], smoothability [29,59], or Varadhan short-time asymptotics for the heat kernel [40].…”