Local polynomial convexity of the union of two totally real surfaces at their intersection

2010/03/25 by Sushil Gorai, Gorai, Sushil
Mathematics · #32E10 #46J10 (Primary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #math.CV #msc:32E10 #msc:46J10

paper · pdf · doi:10.48550/arxiv.1003.4835

18 pages

arxiv created 2010/03/25 · openalex publication_date 2010/03/25 · arxiv updated 2010/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the following question: Let S₁ and S₂ be two smooth, totally-real surfaces in ℂ² that contain the origin. If the union of their tangent planes is locally polynomially convex at the origin, then is S₁ ∪ S₂ locally polynomially convex at the origin? If T₀S₁ ∩ T₀S₂=\0\, then it is a folk result that the answer is yes. We discuss an obstruction to the presumed proof, and provide a different approach. When dimension of T₀S₁ ∩ T₀S₂ over the field of real numbers is 1, we present a geometric condition under which no consistent answer to the above question exists. We then discuss conditions under which we can expect local polynomial convexity.

Related

• On the polynomial convexity of the union of more than two totally-real planes in C²
• Certain real surfaces in ℂ² with isolated singularities
• Certain real surfaces in ℂ² with degenerated CR singularities
• Local Polynomial Convexity at Hyperbolic CR-singularities in M ⊂ ℂⁿ
• Polynomial convexity of compacts that lies in certain Levi-flat hypersurfaces in ℂ²