Embedding of open riemannian manifolds by harmonic functions
Annales de l'Institut Fourier, Volume 25 (1975) no. 1, p. 215-235

Let M be a noncompact Riemannian manifold of dimension n. Then there exists a proper embedding of M into R 2n+1 by harmonic functions on M. It is easy to find 2n+1 harmonic functions which give an embedding. However, it is more difficult to achieve properness. The proof depends on the theorems of Lax-Malgrange and Aronszajn-Cordes in the theory of elliptic equations.

Soit M une variété riemannienne non compacte à n dimensions. Alors il existe un plongement régulier et propre f=(f 1 ,...,f 2n+1 ), de M dans R 2n+1 tel que les fonctions f 1 ,...,f 2n+1 sont harmoniques sur M. Il est facile de trouver 2n+1 fonctions harmoniques qui donnent un plongement régulier. Pour obtenir une telle application qui est à la fois propre, c’est plus subtil. On utilise les théorèmes de Lax-Malgrange et Aronszajn-Cordes dans la théorie d’équations elliptiques.

@article{AIF_1975__25_1_215_0,
     author = {Greene, Robert E. and Wu, H.},
     title = {Embedding of open riemannian manifolds by harmonic functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {25},
     number = {1},
     year = {1975},
     pages = {215-235},
     doi = {10.5802/aif.549},
     zbl = {0307.31003},
     mrnumber = {52 \#3583},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1975__25_1_215_0}
}
Greene, Robert E.; Wu, H. Embedding of open riemannian manifolds by harmonic functions. Annales de l'Institut Fourier, Volume 25 (1975) no. 1, pp. 215-235. doi : 10.5802/aif.549. http://www.numdam.org/item/AIF_1975__25_1_215_0/

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