Maximum modulus sets

Duchamp, Thomas; Stout, Edgar Lee

Annales de l'Institut Fourier, Tome 31 (1981) no. 3, pp. 37-69.

Nous étudions les sous-ensembles du bord d’un domaine strictement pseudoconvexe $D$ de dimension $N$, où la valeur absolue d’une fonction $f$ de $A\left(D\right)$ ou de $A^k\left(D\right)$ prend son maximum. Ces ensembles sont les maximum modulus sets du titre. Si $\Sigma \subset bD$ est une variété différentiable de dimension réelle $N$, et si $\Sigma$ est l’ensemble des points où la valeur absolue d’une fonction $f\in A^2\left(D\right)$ atteint son maximum, alors $\Sigma$ est totalement réelle et elle admet une structure feuilletée avec comme feuilles des variétés compactes qui sont des ensembles pics d’interpolation. Il y a une converse partielle dans le cas analytique réel. Deux fonctions de $A^2\left(D\right)$ qui ont la même variété différentiable de dimension $N$ comme “maximum modulus set”, satisfont une relation analytique, et cette relation est polynomiale si une classe particulière de $H_1(D,\mathbf{R})$ s’annule ou si $\overline{D}\subset {\mathbf{C}}^N$ est polynomialement convexe. Finalement, pour toute fonction $f\in A\left(D\right)$, la dimension topologique de l’ensemble des points où $\left|f\right|$ prend son maximum est au plus $N$.

We investigate some aspects of maximum modulus sets in the boundary of a strictly pseudoconvex domain $D$ of dimension $N$. If $\Sigma \subset bD$ is a smooth manifold of dimension $N$ and a maximum modulus set, then it admits a unique foliation by compact interpolation manifolds. There is a semiglobal converse in the real analytic case. Two functions in $A^2\left(D\right)$ with the same smooth $N$-dimensional maximum modulus set are analytically related and are polynomially related if a certain homology class in $H_1(D,\mathbf{R})$ vanishes or if $\overline{D}\subset {\mathbf{C}}^N$ is polynomially convex. Finally, the maximum modulus set of an arbitrary $f\in A\left(D\right)$ has dimension, in the topological sense, not exceeding $N$.

[1] H. Alexander, Polynomial approximation and hulls in sets of finite linear measure in Cn, Amer. J. Math., 93 (1971), 65-74. | MR | Zbl

[2] A. Andreotti and R. Narasimhan, A topological property of Runge pairs, Ann. Math., (2) 76 (1962), 499-509. | MR | Zbl

[3] E. Bishop, A generalization of the Stone-Weierstrass theorem, Pacific J. Math., 11 (1961), 777-783. | MR | Zbl

[4] D.E. Blair, Contact Manifolds in Riemannian Geometry, Springer Lecture Notes in Mathematics, vol. 509, Springer-Verlag, Berlin, Heidelberg, New York, 1976. | MR | Zbl

[5] A. Browder, Cohomology of maximal ideal spaces, Bull. Amer. Math. Soc., 67 (1961), 515-516. | MR | Zbl

[6] D. Burns and E.L. Stout, Extending functions from submanifolds of the boundary, Duke Math., J., 43 (1976), 391-404. | MR | Zbl

[7] H. Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France, 85 (1957), 77-99. | Numdam | MR | Zbl

[8] J. Chaumat and A.M. Chollet, Ensembles pics pour A∞ (D), Ann. Inst. Fourier, Grenoble, XXIX (1979), 171-200. | Numdam | MR | Zbl

[9] A.M. Davie and B. Øksendal, Peak interpolation sets for some algebras of analytic functions, Pacific J. Math., 41 (1972), 81-87. | MR | Zbl

[10] H. Federer, Geometric Measure Theory, Springer-Verlag New York, Inc., New York, 1969. | MR | Zbl

[11] T. Duchamp, The classification of Legendre embeddings, to appear.

[12] J.E. Fornaess, Embedding strictly pseudoconvex domains in convex domains, Amer. J. Math., 98 (1976), 529-569. | MR | Zbl

[13] R. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. | MR | Zbl

[14] C.D. Hill and G. Taiani, Families of analytic discs in Cn with boundaries on a prescribed CR submanifold, Ann. Scuola Norm. Sup. Pisa Sci., (IV) V, (1978), 327-380. | Numdam | MR | Zbl

[15] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, 1962. | MR | Zbl

[16] H. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, Princeton, 1948. | Zbl

[17] V.S. Klein, Behavior of Holomorphic Functions at Generating Submanifolds of the Boundary, doctoral dissertation, University of Washington, Seattle, 1979.

[18] H.B. Lawson, Lectures on the Quantitative Theory of Foliations, CBMS Regional Conference Series in Mathematics, Number 27, American Mathematical Society, Providence, Rhode Island, 1977. | Zbl

[19] L. Loomis and S. Sternberg, Advanced Calculus, Addison-Wesley, Reading, 1968. | MR | Zbl

[20] J. Milnor, Topology from the Differentiable Viewpoint, University Press of Virginia, Charlottesville, 1965. | MR | Zbl

[21] M. Müller, Geometrisch Untersuchungen allgemeiner und einiger spezieller Pseudokonvexer Gebiete, Bonner Math. Schriften, 78, Bonn, 1975. | Zbl

[22] S.I. Pinchuk, A boundary uniqueness theorem for holomorphic functions of several complex variables, Math. Notes, 15 (1974), 116-120. | MR | Zbl

[23] M. Range and Y.-T. Siu, Ck approximation by holomorphic functions and $ATT$-closed forms on Ck submanifolds of a complex manifold, Math. Ann., 210 (1974), 105-122. | Zbl

[24] G. Reeb, Sur certaines propriétés topologiques des variétés feuilletées, Act. Sci. Indust., 1183, Hermann, Paris, 1952. | MR | Zbl

[25] W. Rudin, Peak interpolation manifolds of class C1, Pacific J. Math., 75 (1978), 267-279. | MR | Zbl

[26] W. Rudin, Lectures on the Edge-of-the-Wedge Theorem, CBMS Regional Conference Series in Mathematics, Number 6, American Mathematical Society, Providence, Rhode Island, 1971. | MR | Zbl

[27] W. Rudin and E.L. Stout, Boundary properties of functions of several complex variables, J. Math. Mech., 14 (1965), 991-1006. | MR | Zbl

[28] A. Sadullaev, A boundary uniqueness theorem in Cn, Math. USSR Sbornik, 30 (1976), 501-514. | Zbl

[29] J. Schwartz, Nonlinear Functional Analysis, Gordon and Breach, New York, 1969. | MR | Zbl

[30] B. Shiffman, On the continuation of analytic curves, Math. Ann., 184 (1970), 268-274. | MR | Zbl

[31] N. Sibony, Valeurs au bord de fonctions holomorphes et ensembles polynomialement convexes, Séminaire Pierre Lelong 1975-1976. Springer Lecture Notes in Mathematics, vol. 578, Springer-Verlag, Berlin, Heidelberg, New York, 1977. | Zbl

[32] K. Stein, Analytische Projektion komplexer Mannigfaltigkeiten, Colloque sur les Fonctions de Plusieurs Variables, Brussels, 1953. George Throne, Leige and Masson, Paris, 1953. | Zbl

[32a] K. Stein, Die Existenz Komplexer Basen zu holomorphen Abbildungen, Math. Ann., 136 (1958), 1-8. | MR | Zbl

[33] S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, 1964. | MR | Zbl

[34] E.L. Stout, The Theory of Uniform Algebras, Bogden and Quigley, Tarrytown-on-Hudson and Belmont, 1971. | MR | Zbl

[35] E.L. Stout, Interpolation manifolds, Recent Developments in Several Complex Variables, Annals of Mathematics Studies, to appear. | Zbl

[36] A.E. Tumanov, A peak set for the disc algebra of metric dimension 2.5 in the three-dimensional unit sphere, Math. USSR Izvestija, 11 (1977), 370-377. | MR | Zbl

[37] B.M. Weinstock, Zero-sets of continuous holomorphic functions on the boundary of a strongly pseudoconvex domain, J. London Math. Soc., 18 (1978), 484-488. | MR | Zbl

[38] R.O. Wells, Compact real submanifolds of a complex manifold with nondegenerate holomorphic tangent bundles, Math. Ann., 179 (1969), 123-129. | MR | Zbl

[39] R.O. Wells, Real analytic subvarieties and holomorphic approximation, Math. Ann., 179 (1969), 130-141. | MR | Zbl

[40] A. Zygmund, Trigonometric Series, vol. I., Cambridge University Press, Cambridge, 1959. | Zbl