Certain real surfaces in ℂ 2 with isolated singularities

The surfaces in ℂ 2 with an isolated CR singularity at the origin and with cubic lowest degree homogeneous term in its graph near the origin, under certain geometric condition, can be reduced — up to biholomorphism of ℂ 2 — to a one-parameter family of the form M t : = ( z , w ) ∈ ℂ 2 : w = z 2 z ¯ + t z z ¯ 2 + t 2 3 z ¯ 3 + o | z | 3 , t ∈ ( 0 , ∞ ) , near the origin. We prove that M t is not locally polynomially convex if t < 1 . The local hull contains a ball centred at the origin if t < 3 / 2 . We also prove that M t is locally polynomially convex for t ≥ 3 / 2 . We show that, for 3 / 2 ≤ t < 1 , the local hull of M t contains a one-parameter family of analytic discs passing through the origin. We also prove that, if we remove the higher order terms from the graphing function of M t , it is locally polynomially convex for t ≥ 15 - 33 2 2 . Some new results about the local polynomial convexity of the union of three totally-real planes are also reported.