A note on common zeroes of Laplace--Beltrami eigenfunctions

Details A note on common zeroes of Laplace--Beltrami eigenfunctions A note on common zeroes of Laplace--Beltrami eigenfunctions

2005-11-28

arxiv.org/abs/math/0511688v1

Annals of Global Anal. and Geom., 26 (2004), 201-208

Mathematics

Metric Geometry

8 pages. This is the published paper with several additional comments in footnotes

Scientific paper

Let $\De u+\la u=\De v+\la v=0$, where $\De$ is the Laplace--Beltrami
operator on a compact connected smooth manifold $M$ and $\la>0$. If $H^1(M)=0$
then there exists $p\in M$ such that $u(p)=v(p)=0$. For homogeneous $M$,
$H^1(M)\neq0$ implies the existence of a pair $u,v$ as above that has no common
zero.

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