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POLYNOMIAL CONVEXITY OF COMPACTS THAT LIES IN CERTAIN LEVI-FLAT HYPERSURFACES IN $\mathbb {C}^2$

Part of: Holomorphic convexity

Published online by Cambridge University Press:  07 July 2025

SUSHIL GORAI Open the ORCID record for SUSHIL GORAI [Opens in a new window]  and
GOLAM MOSTAFA MONDAL Open the ORCID record for GOLAM MOSTAFA MONDAL [Opens in a new window]
SUSHIL GORAI*
Affiliation:
Department of Mathematics and Statistics https://ror.org/00djv2c17Indian Institute of Science Education and Research Kolkata Mohanpur – 741 246 India
GOLAM MOSTAFA MONDAL
Affiliation:
Department of Mathematics https://ror.org/04dese585 Indian Institute of Science Bangalore – 560012 India golammostafaa@gmail.com golammondal@iisc.ac.in
*
sushil.gorai@gmail.com sushil.gorai@iiserkol.ac.in
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Abstract

In this paper, we first prove that the totally real discs lying in certain Levi-flat hypersurfaces are polynomially convex. We also studied the polynomial convexity of totally real discs lying in the regular part of certain singular Levi-flat hypersurfaces. In particular, a necessary and sufficient condition for polynomial convexity of totally real discs lying in the non-singular part of the boundary of the Hartogs triangle is achieved. Sufficient conditions on general compact subsets lying on those hypersurfaces for polynomial convexity are also reported here.

Keywords

polynomial convexitytotally real discLevi-flat hypersurfaces

MSC classification

Primary: 32E20: Polynomial convexity

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Type
Article
Information
Nagoya Mathematical Journal , First View , pp. 1 - 13
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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References

Alexander, H., Totally real sets in ${\mathbb{C}}^2$ , Proc. Amer. Math. Soc. 111 (1991), 131–133.Google Scholar
Burns, D. and Gong, X., Singular Levi-flat real analytic hypersurfaces , Amer. J. Math. 121 (1999), 23–53.CrossRefGoogle Scholar
Duval, J., Un exemple de disque polynômialement convexe , Math. Ann. 281 (1988), 583–588.CrossRefGoogle Scholar
Duval, J. and Lawrence, M., Polynomial hull of a torus fibered over the circle , Int. J. Math. 29 (2018), Art. ID 1850031, 10 pp.CrossRefGoogle Scholar
Forstnerič, F., Polynomial hulls of sets fibered over the circle , Indiana Univ. Math. J. 37 (1988), 869–889.CrossRefGoogle Scholar
Forstnerič, F. and Laurent-Thiébaut, C., Stein compacts in Levi-flat hypersurfaces , Trans. Amer. Math. Soc. 360 (2008), 307–329.CrossRefGoogle Scholar
Gorai, S., A note on polynomial convexity of the union of finitely many totally-real planes in ${\mathbb{C}}^2$ , J. Math. Anal. Appl. 418 (2014), 842–851.CrossRefGoogle Scholar
Gorai, S. and Mondal, G. M., Uniform approximation on certain polynomial polyhedra in ${\mathbb{C}}^2$ , to appear in Trans. Amer. Math. Soc. 378 (2025) no. 8, 5441–5473.Google Scholar
Havin, V. P. and Nikolski, N. K., Linear and Complex Analysis Problem Book 3: Part 2, Lecture Notes in Mathematics, Springer, Berlin, 2006.Google Scholar
Hörmander, L. and Wermer, J., Uniform approximation on compact sets in ${\mathbb{C}}^n$ , Math. Scand. 23 (1968), 5–21.CrossRefGoogle Scholar
Jöricke, B., Removable singularities of CR-functions , Ark. Mat. 26 (1988), 117–143.CrossRefGoogle Scholar
Nirenberg, R. and Wells, R. O. Jr., Approximation theorems on differentiable submanifolds of a complex manifold , Trans. Amer. Math. Soc. 142 (1969), 15–35.CrossRefGoogle Scholar
O’Farrell, A. G. and Preskenis, K. J., Approximation by polynomials in two diffeomorphisms , Bull. Amer. Math. Soc. (N.S.) 10 (1984), 105–107.CrossRefGoogle Scholar
O’Farrell, A. G. and Preskenis, K. J., Uniform approximation by polynomials in two functions , Math. Ann. 284 (1989), 529–535.CrossRefGoogle Scholar
Rudin, W., Functional Analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, NY, 1991.Google Scholar
Samuelsson, H. and Wold, E. F., Uniform algebras and approximation on manifolds , Invent. Math. 188 (2012), 505–523.CrossRefGoogle Scholar
Słodkowski, Z., Polynomially convex hulls of families of arcs , Studia Math. 165 (2004), 1–17.CrossRefGoogle Scholar
Słodkowski, Z., Holomorphic motions and polynomial hulls , Proc. Amer. Math. Soc. 111 (1991), 347–355.CrossRefGoogle Scholar
Stolzenberg, G., Polynomially and rationally convex sets , Acta Math. 109 (1963), 259–289.CrossRefGoogle Scholar
Stout, E. L., “Removable singularities for the boundary values of holomorphic functions” in J. E. Fornaess (ed.) Several Complex Variables (Stockholm, 1987/1988), Mathematical Notes, 38, Princeton University Press, Princeton, NJ, 1993, 600–629.Google Scholar
Stout, E. L., Polynomial Convexity, Progress in Mathematics, 261, Birkhäuser Boston, Inc., Boston, MA, 2007.Google Scholar
Wermer, J., Approximation on a disk , Math. Ann. 155 (1964), 331–333.CrossRefGoogle Scholar