Embedding of open riemannian manifolds by harmonic functions

Greene, Robert E.; Wu, H.

Annales de l'Institut Fourier, Tome 25 (1975) no. 1, pp. 215-235.

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 ${\mathbf{R}}^{2n+1}$ 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.

Let $M$ be a noncompact Riemannian manifold of dimension $n$. Then there exists a proper embedding of $M$ into ${\mathbf{R}}^{2n+1}$ 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.

[1] N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl., 36 (1957), 235-249. | MR | Zbl

[2] S. Bochner, Analytic mappings of compact Riemann spaces into Euclidean space, Duke Math. J., 3 (1937), 339-354. | JFM | Zbl

[3] S.S. Cairns, On the triangulation of regular loci, Ann. of Math., 35 (1934), 579-587. | JFM | Zbl

[4] S.-S. Chern, An elementary proof of the existence of isothermal parameters on a surface, Proc. Amer. Math. Soc., 6 (1955), 771-782. | MR | Zbl

[5] H.O. Cordes, Über die Bestimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorgaben, Nachr. Akad. Wiss. Göttingen, Math. Phys. K1. IIa, no 11 (1956), 239-258. | MR | Zbl

[6] W. Feller, Über die Lösungen der linearen partiellen Differentialgleichungen zweiter Ordnung von elliptischen Typus, Math. Ann., 102 (1930), 633-649. | JFM

[7] H. Grauert, On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math., 68 (1958), 460-472. | MR | Zbl

[8] R.E. Greene, Isometric Embedding of Riemannian and Pseudo-Riemannian Manifolds, Memoir 97, Amer. Math. Soc., 1970. | MR | Zbl

[9] R.E. Greene, and H. Wu, Integrals of subharmonic functions on manifolds of nonnegative curvature (to appear). | Zbl

[10] L. Hörmander, An Introduction to Complex Analysis in Several variables. D. Van Nostrand, Princeton, New Jersey, 1966. | Zbl

[11] P.D. Lax, A stability theorem for abstract differential equations and its application to the study of the local behavior of solutions of elliptic equations, Comm. Pur Appl. Math., 9 (1956), 747-766. | MR | Zbl

[12] B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Annales de l'Institut Fourier, 6 (1955-1956), 271-355. | Numdam | MR | Zbl

[13] C.B. Morrey, The analytic imbedding of abstract real-analytic manifolds, Ann. of Math., 68 (1958), 159-201. | MR | Zbl

[14] J.R. Munkres, Elementary Differential Topology, Princeton University Press, Princeton, New Jersey, 1966.

[15] R. Narasimhan, Analysis on Real and Complex Manifolds, North-Holland, Amsterdam, 1968. | MR | Zbl

[16] J.F. Nash, The imbedding problem for Riemannian manifolds, Ann. of Math., 63 (1956), 20-63. | MR | Zbl

[17] A. Pliś, A smooth linear elliptic differential equation without a solution in a sphere, Comm. Pure Appl. Math., 14 (1961), 599-617. | MR | Zbl

[18] M.H. Protter, Unique continuation for elliptic equations, Trans. Amer. Math. Soc., 95 (1960), 81-91. | MR | Zbl

[19] G. De Rham, Variétés différentiables, Hermann, Paris, 1955. | Zbl

[20] H.L. Royden, The analytic approximation of differentiable mappings, Math. Ann., 139 (1960), 171-179. | MR | Zbl

[21] H. Whitney, Analytic coordinate systems and arcs in a manifold, Ann. of Math., 38 (1937), 809-818. | JFM | Zbl

[22] H. Wu, Remarks on the first main theorem in equidistribution theory. II, J. Differential Geometry, 2 (1968), 369-384. | MR | Zbl

[23] L. Bers, Local behavior of solutions of general linear elliptic equations, Comm. Pure Appl. Math., 8 (1955), 473-496. | MR | Zbl

[24] L. Bers, F. John, and M. Schechter, Partial Differential Equations, Interscience Publishers, New York, 1964.

[25] B. Malgrange, Plongement des variétés analytiques-réelles, Bull. Soc. Math. France, 85 (1957), 101-113. | Numdam | MR | Zbl