Rational approximation and Lagrangian inclusions

2015/04/08 by Shafikov, Rasul, Sukhov, Alexandre
#32E20 #32E30 #32V40 #53D12 #Complex Variables (math.CV) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1504.02083

Abstract

We show that a Lagrangian inclusion in \mathbb C² with double transverse self-intersection points and standard open Whitney umbrellas is rationally convex. As an application we show that any compact surface S, except S² and \mathbb RP₂, admits a pair of smooth complex-valued functions f₁, f₂ with the property that any continuous complex valued function on S is a uniform limit of a sequence of R_j(f₁,f₂), where R_j(z₁,z₂) are rational functions in \mathbb C².

Related

• Lagrangian inclusion with an open Whitney umbrella is rationally convex
• Rationally convex embeddings of real surfaces into ℂ²
• A complex-analytic characterization of Lagrangian immersions in \mathbb Cⁿ with transverse double points
• Local polynomial convexity of the unfolded Whitney umbrella in \mathbb C²
• Rational convexity of non generic immersed lagrangian submanifolds