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A note on common zeroes of Laplace--Beltrami eigenfunctions

V. M. Gichev
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Abstract

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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V. M. Gichev. 2005-11-28. A note on common zeroes of Laplace--Beltrami eigenfunctions. https://arxiv.org/abs/math/0511688

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