  We investigate plasmon resonances for curved nanorods which present
anisotropic geometries. We analyze quantitative properties of the plasmon
resonance and its relationship to the metamaterial configurations and the
anisotropic geometries of the nanorods. Based on delicate and subtle asymptotic
and spectral analysis of the layer potential operators, particularly the
Neumann-Poincar\'e operators, associated with anisotropic geometries, we derive
sharp asymptotic formulae of the corresponding scattering field in the
quasi-static regime. By carefully analyzing the asymptotic formulae, we
establish sharp conditions that can ensure the occurrence of the plasmonic
resonance. The resonance conditions couple the metamaterial parameters, the
wave frequency and the nanorod geometry in an intricate but elegant manner. We
provide thorough resonance analysis by studying the wave fields both inside and
outside the nanorod. Furthermore, our quantitative analysis indicates that
different parts of the nanorod induce varying degrees of resonance.
Specifically, the resonant strength at the two end-parts of the curved nanorod
is more outstanding than that of the facade-part of the nanorod. This paper
presents the first theoretical study on plasmon resonances for nanostructures
within anisotropic geometries.
