# Splitting and gap theorems in the presence of a Poincaré-Sobolev inequality

Chapter

## Abstract

Up to now, we have been using Theorem 4.5 to show that solutions of a differential problem of the type have to be identically zero. The aim of this section is to present a geometrical problem in which the second alternative of Theorem 4.5 does actually occur, that is,
.

$$
\left\{ \begin{gathered}
\psi \Delta \psi + a\left( x \right)\psi ^2 \geqslant - A\left| {\left. {\nabla \psi } \right|^2 ,} \right. \hfill \\
\psi \geqslant 0 \hfill \\
\end{gathered} \right.
$$

*ψ*becomes a positive solution of the linear equation$$
\Delta \psi + a\left( x \right)\psi = 0
$$

## Keywords

Riemannian Manifold Scalar Curvature Sobolev Inequality Integral Curve Ricci Tensor
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