Open Whitney umbrellas are locally polynomially convex

Mitrea, Octavian; Shafikov, Rasul

doi:10.1090/proc/13251

Open Whitney umbrellas are locally polynomially convex
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by Octavian Mitrea and Rasul Shafikov PDF

Proc. Amer. Math. Soc. 144 (2016), 5319-5332 Request permission

Abstract:

It is proved that any smooth open Whitney umbrella in $\mathbb {C}^2$ is locally polynomially convex near the singular point.

References

John T. Anderson, Alexander J. Izzo, and John Wermer, Polynomial approximation on real-analytic varieties in $\mathbf C^n$, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1495–1500. MR 2053357, DOI 10.1090/S0002-9939-03-07263-0
Michèle Audin, Quelques remarques sur les surfaces lagrangiennes de Givental′, J. Geom. Phys. 7 (1990), no. 4, 583–598 (1991) (French, with English summary). MR 1131914, DOI 10.1016/0393-0440(90)90008-Q
Marco Brunella and Massimo Miari, Topological equivalence of a plane vector field with its principal part defined through Newton polyhedra, J. Differential Equations 85 (1990), no. 2, 338–366. MR 1054553, DOI 10.1016/0022-0396(90)90120-E
Alexander D. Bruno, Local methods in nonlinear differential equations, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1989. Part I. The local method of nonlinear analysis of differential equations. Part II. The sets of analyticity of a normalizing transformation; Translated from the Russian by William Hovingh and Courtney S. Coleman; With an introduction by Stephen Wiggins. MR 993771, DOI 10.1007/978-3-642-61314-2
Freddy Dumortier, Singularities of vector fields on the plane, J. Differential Equations 23 (1977), no. 1, 53–106. MR 650816, DOI 10.1016/0022-0396(77)90136-X
Julien Duval, Surfaces convexes dans un bord pseudoconvexe, Astérisque 217 (1993), 6, 103–118 (French, with French summary). Colloque d’Analyse Complexe et Géométrie (Marseille, 1992). MR 1247754
A. B. Givental′, Lagrangian imbeddings of surfaces and the open Whitney umbrella, Funktsional. Anal. i Prilozhen. 20 (1986), no. 3, 35–41, 96 (Russian). MR 868559
Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
B. Jöricke, Removable singularities of CR-functions, Ark. Mat. 26 (1988), no. 1, 117–143. MR 948284, DOI 10.1007/BF02386112
S. Nemirovski, private communication, 2010.
Stefan Nemirovski and Kyler Siegel, Rationally convex domains and singular Lagrangian surfaces in $\Bbb {C}^2$, Invent. Math. 203 (2016), no. 1, 333–358. MR 3437874, DOI 10.1007/s00222-015-0598-4
Hugo Rossi, The local maximum modulus principle, Ann. of Math. (2) 72 (1960), 1–11. MR 117539, DOI 10.2307/1970145
Rasul Shafikov and Alexandre Sukhov, Local polynomial convexity of the unfolded Whitney umbrella in $\Bbb C^2$, Int. Math. Res. Not. IMRN 22 (2013), 5148–5195. MR 3129096, DOI 10.1093/imrn/rns199
Rasul Shafikov and Alexandre Sukhov, Polynomially convex hulls of singular real manifolds, Trans. Amer. Math. Soc. 368 (2016), no. 4, 2469–2496. MR 3449245, DOI 10.1090/tran/6422
Edgar Lee Stout, Polynomial convexity, Progress in Mathematics, vol. 261, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2305474, DOI 10.1007/978-0-8176-4538-0