Some generalizations of Calabi compactness theorem

TL;DR

This paper extends Calabi's compactness theorem to include cases with negative Ricci curvature, providing new criteria for manifold properties and analyzing solutions of a related differential equation with a unified approach.

Contribution

It introduces generalized compactness criteria for Riemannian manifolds with negative Ricci curvature using a unified ODE approach based on critical curves.

Findings

New criteria are sharp and improve previous results in borderline cases.

Criteria apply to the positivity, zero existence, and oscillatory behavior of solutions to a specific ODE.

The approach unifies various cases through the concept of critical curves.

Abstract

In this paper we obtain generalized Calabi-type compactness criteria for complete Riemannian manifolds that allow the presence of negative amounts of Ricci curvature. These, in turn, can be rephrased as new conditions for the positivity, for the existence of a first zero and for the nonoscillatory-oscillatory behaviour of a solution $g(t)$ of $g"+Kg=0$, subjected to the initial condition $g(0)=0$, $g'(0)=1$. A unified approach for this ODE, based on the notion of critical curve, is presented. With the aid of suitable examples, we show that our new criteria are sharp and, even for $K\ge 0$, in borderline cases they improve on previous works of Calabi, Hille-Nehari and Moore.