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

Part of the Progress in Mathematics book series (PM, volume 266)


Up to now, we have been using Theorem 4.5 to show that solutions of a differential problem of the type
$$ \left\{ \begin{gathered} \psi \Delta \psi + a\left( x \right)\psi ^2 \geqslant - A\left| {\left. {\nabla \psi } \right|^2 ,} \right. \hfill \\ \psi \geqslant 0 \hfill \\ \end{gathered} \right. $$
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, ψ becomes a positive solution of the linear equation
$$ \Delta \psi + a\left( x \right)\psi = 0 $$


Riemannian Manifold Scalar Curvature Sobolev Inequality Integral Curve Ricci Tensor 
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© Birkhäuser Verlag AG 2008

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