Abstract
Let Δu+λu=Δv+λv= 0, where Δ isthe Laplace–Beltrami operator on a compact connected smoothmanifold M and λ > 0. If H 1(M) = 0then there exists p∈M such that u(p)=v(p) = 0 For homogeneous M,H 1(M) ≠ 0 implies the existence of a pair u,v as above that has no common zero.
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Gichev, V.M. A Note on Common Zeroes of Laplace–Beltrami Eigenfunctions. Annals of Global Analysis and Geometry 26, 201–208 (2004). https://doi.org/10.1023/B:AGAG.0000031091.61435.a5
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DOI: https://doi.org/10.1023/B:AGAG.0000031091.61435.a5