Properties of surface helicity and its optimisation

In [J. Cantarella, J. Parsley, J. Geom. Phys. 60:1127 (2010)] Cantarella and Parsley introduced the notion of submanifold helicity. In the present paper, we investigate properties of surface helicity and, in particular, answer two open questions posed in the aforementioned work: (i) We give a precise mathematically rigorous physical interpretation of surface helicity in terms of linking of distinct field lines, and (ii) prove that surface helicity is non-trivial if and only if the underlying surface has non-trivial topology (i.e., at least one hole).We then focus on toroidal surfaces and consider the problem of optimising surface helicity among surfaces of fixed area. We show that rotationally symmetric toroidal surfaces constitute global minimisers.