Certain real surfaces in $\mathbb{C}^2$ with isolated singularities

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.