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

Abstract

We consider the following question: Let S1 and S2 be two smooth, totally-real surfaces in C 2 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 dim R (T0S1 ∩T0S2) = 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.2000 Mathematics Subject Classification. Primary: 32E20, 46J10.

On regular Stein neighborhoods of a union of two maximally totally real subspaces in $$\mathbb {C}^n$$ C n

“…At the end we also give the application of this result to totally real immersions of real n-manifolds in C n with only finitely many double points, and such that the union of the tangent spaces at each intersection in some local coordinates coincides with M (A) ∪ N , described above. In connection to this we also note that Weinstock's result has been recently generalised by Gorai [6] and Shafikov and Sukhov [12,Theorems 1.3 and 4.2], to the effect that a union of two maximally totally real submanifolds in C n , intersecting transversally at the origin, is polynomially convex near the origin, if the union of their tangent spaces at the origin is polinomially convex near the origin.…”

Polynomially convex hulls of singular real manifolds

“…Gorai [26] proved this for n = 2. Also note that Theorem 1.3 is an immediate consequence of Theorem 4.2 and Lemma 4.1.…”

“…Case 4. Consider the Jordan block of size k given by (26) corresponding to a complex eigenvalue s j + it j of A. As it was mentioned previously, we always have |t j | < 1 if s j = 0.…”

Certain smooth real surfaces in $\mathbb{C}^2$ with singularity

“…It is not trivial to pass the local polynomial convexity at the origin to the union of totally-real submanifolds from the union of their tangent spaces at the origin. Union of two totally-real submanifolds in C n intersection transversely at the origin is locally polynomially convex if the union of their tangent spaces is [10,24]. These results uses Weinstock's result (Result 2.6) crucially.…”