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

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.

### MSC:

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

### Keywords:

pinching; Schrödinger operator; comparison
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