A finite-dimensionality result

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


As briefly mentioned at the beginning of the previous chapter, typical geometric applications of Theorem 4.5 are obtained by applying it when the function ψ is the norm of the section of a suitable vector bundle. In appropriate circumstances, the theorem guarantees that certain vector subspaces of such sections are trivial, the main geometric assumption being the existence of a positive solution ϕ of the differential inequality
$$ \Delta \varphi + Ha\left( x \right)\varphi \leqslant 0 weakly on M, $$
where a(x) is a lower bound for the relevant curvature term. According to Lemma 3.10 this amounts to requiring that the bottom of the spectrum of the Schrödinger operator −Δ − Ha(x) is non-negative.


Riemannian Manifold Vector Bundle Harnack Inequality Morse Index Complete Riemannian Manifold 
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© Birkhäuser Verlag AG 2008

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