  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.
