Part of the Progress in Mathematics book series (PM, volume 266)
A finite-dimensionality result
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
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.
$$ \Delta \varphi + Ha\left( x \right)\varphi \leqslant 0 weakly on M, $$
KeywordsRiemannian Manifold Vector Bundle Harnack Inequality Morse Index Complete Riemannian Manifold
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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