\(L^ p\)-pinching and the geometry of compact Riemannian manifolds. (English) Zbl 0821.53033

Le Couturier, Michel; Robert, Gilles F.

Comment. Math. Helv. 69, No. 2, 249-271 (1994).

The authors prove a Harnack-type inequality for sections of Riemannian vector bundle over a compact manifold, lying in the kernel of a Schrödinger operator under \(L^ p\)-pinching assumptions on the potential.
Many geometric applications are given generalizing for example classical comparison results due to Bochner to the case of almost non-negative resp. non-positive Ricci curvature (in the sense of \(L^ p\)-estimates). Another application is that the minimal volume of an almost nonpositively curved manifold with infinite isometry group vanishes.

Reviewer: H.-B.Rademacher (Leipzig)

MSC:

53C20	Global Riemannian geometry, including pinching
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Keywords:

pinching; Schrödinger operator; comparison