Hulls of deformations in $\textbf {C}^{n}$

Abstract:

A problem of ${\text {E}}$. Bishop on the polynomially convex hulls of deformations of the torus is considered. Let the torus ${T^2}$ be the distinguished boundary of the unit polydisc in ${{\mathbf {C}}^2}$. If $t \mapsto T_t^2$ is a smooth deformation of ${T^2}$ in ${{\mathbf {C}}^2}$ and ${g_0}$ is an analytic disc in ${{\mathbf {C}}^2}$ with boundary in ${T^2}$, a smooth family of analytic discs $t \mapsto {g_t}$, is constructed with the property that the boundary of ${g_t}$ lies in $T_t^2$. This construction has implications for the polynomially convex hulls of the tori $T_t^2$. An analogous problem for a $2$-sphere in ${{\mathbf {C}}^2}$ is also considered.