In this survey article, we present two applications of surface curvatures in
theoretical physics. The first application arises from biophysics in the study of the shape of
cell vesicles involving the minimization of a mean curvature type energy called the Helfrich
bending energy. In this formalism, the equilibrium shape of a cell vesicle may present itself
in a rich variety of geometric and topological characteristics. We first show that there is
an obstruction, arising from the spontaneous curvature, to the existence of a minimizer of
the Helfrich energy over the set of embedded ring tori. We then propose a scale-invariant
anisotropic bending energy, which extends the Canham energy, and show that it possesses
a unique toroidal energy minimizer, up to rescaling, in all parameter regime. Furthermore,
we establish some genus-dependent topological lower and upper bounds, which are known
to be lacking with the Helfrich energy, for the proposed energy. We also present the
shape equation in our context, which extends the Helfrich shape equation. The second
application arises from astrophysics in the search for a mechanism for matter accretion in
the early universe in the context of cosmic strings. In this formalism, gravitation may
simply be stored over a two-surface so that the Einstein tensor is given in terms of the
Gauss curvature of the surface which relates itself directly to the Hamiltonian energy
density of the matter sector. This setting provides a lucid exhibition of the interplay of
the underlying geometry, matter energy, and topological characterization of the system.
In both areas of applications, we encounter highly challenging nonlinear partial differential
equation problems. We demonstrate that studies on these equations help us to gain
understanding of the theoretical physics problems considered.
