Abstract
Suppose that the Laplace operator of a complete Riemannian manifold admits two linearly independent real-valued eigenfunctions, which do not vanish simultaneously. Consider the complex-valued function with these functions as the real and imaginary parts. This last function never vanishes and defines a homotopically non-trivial map into the punctured complex plane.
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Donnelly, H. Winding numbers and eigenfunctions of the Laplacian. Ann Glob Anal Geom 44, 1–3 (2013). https://doi.org/10.1007/s10455-012-9352-y
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DOI: https://doi.org/10.1007/s10455-012-9352-y