Under certain geometric condition, the surfaces in $\mathbb{ C}^2$ with isolated CR singularity at the origin and with cubic lowest degree homogeneous term in its graph near the origin, can be reduced, up to biholomorphism of $\mathbb{ C}^2$, to a one parameter family of the form \[ M_t:=\left\{ (z,w)\in\mathbb{ C}^2: w=z^2\overline{ z}+tz\overline{ z}^2+\dfrac{ t^2}{ 3} \overline{ z}^3+o(|z|^3)\right\},\;\; t\in (0,\infty) \] 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 $t<\sqrt{ 3}/2$. We also prove that $M_t$ is locally polynomially convex for $t\geq\sqrt{ \dfrac{ 3}{ 2}}$. We show that, for $\sqrt{ 3}/2\leq t<1$, the polynomial hull of $M_t\cap \overline{ B(0;\delta)}$ contains a one parameter family of analytic discs passing through the origin for every $\delta>0$. We also prove that, if we remove the higher order terms from the graphing function of $M_t$, it is locally polynomially convex for $t\geq\dfrac{ \sqrt{ 15-\sqrt{ 33}}}{ 2\sqrt{ 2}}$. Some new results about the local polynomial convexity of the union of three totally-real planes are also reported.
Categories: math.CV