Certain real surfaces in ℂ² with degenerated CR singularities

2026/07/19 by Sushil Gorai, Suman Karak, Golam Mostafa Mondal
Mathematics · #math.CV

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Abstract

In this paper, we study the local polynomial convexity of certain smooth real surfaces in \(ℂ²\) with isolated CR singularity at the origin with higher-order of degeneracy. Under the assumption that the surface can be pulled back to a union of finitely many pairwise transverse totally real surfaces by a proper holomorphic map from ℂ² to ℂ², we obtain a normal form for such surfaces near the origin as \(z,w)∈ℂ²: w= z^k+o(|z|^k)\ or M_t := \ (z,w)∈ℂ² : w=(z+tz)^k+o(|z|^k) \, for some \(t>0\), where the parameter t is a local biholomorphic invariant. We focus on the surfaces with order of degeneracy k≥ 3. We prove that M_t is locally polynomially convex at the origin if t>cosec(\fracπk). On the other hand, for 0<t<(1)/(k-2), we will also show that M_t fails to be locally polynomially convex at the origin; and furthermore, a (2k-3)-parameter family of analytic discs attached to M_t for 0<t<min\sin(\fracπk),(1)/(k-2)\.

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