A toroidal tube solution to a problem involving mean curvature and Newtonian potential

Abstract

The Ohta–Kawasaki theory for block copolymer morphology and the Gierer–Meinhardt theory for morphogenesis in cell development both give rise to a nonlocal geometric problem. One seeks a set in $R^3$ which satisfies an equation that links the mean curvature of the boundary of the set to the Newtonian potential of the set. An axisymmetric, torus shaped, tube like solution exists in $R^3$ if the lone parameter of the problem is sufficiently large. A cross section of the torus is small and the distance from the center of the cross section to the axis of symmetry is large. The solution is stable in the class of axisymmetric sets in a particular sense. This work is motivated by the recent discovery of a toroidal morphological phase in a triblock copolymer.