Surface plasmon polaritons propagating along curved metal-dielectric interfaces experience geometry-induced modifications absent on flat surfaces. In this work, we derive a covariant, effective two-dimensional wave equation for the transverse magnetic surface plasmon mode on weakly curved smooth interfaces. By perturbatively expanding Maxwell's equations with curvature-adapted boundary conditions, we find a Helmholtz equation with two geometric potential terms that enter at first order in the extrinsic curvature: an isotropic contribution proportional to the extrinsic curvature, and an anisotropic operator arising from the traceless part of the second fundamental form. These linear-in-curvature potentials distinguish convex from concave interfaces, in contrast to the quadratic potentials known from symmetrically confined systems such as dielectric waveguides. We show that our equation reproduces established results for spherical and cylindrical interfaces. We furthermore predict that the anisotropic contribution vanishes when the ratio of the material permittivities equals the square of the golden ratio. As an application, we demonstrate sign-dependent cooperative frequency shifts as well as a curvature-driven redistribution of superradiant and subradiant decay rates for a ring of quantum emitters on a curved metallic spheroid interacting through the surface plasmons.

Spontaneous symmetry breaking leads to diverse phenomena across the natural sciences, from the Higgs mechanism in particle physics to superconductors and collective animal behavior. In photonic systems, the symmetry of light states can be broken when two optical fields interact through the Kerr nonlinearity, as shown in early demonstrations with counterpropagating and cross-polarized modes. Here, we report the first observation of color symmetry breaking in an integrated silicon nitride microring, where spontaneous power imbalance arises between optical mode at different wavelengths, mediated by the Kerr effect. The threshold power for this effect is as low as 19 mW. By examining the system's homogeneous states, we further demonstrate a Kerr-based nonlinear activation-function generator that produces sigmoid-, quadratic-, and leaky-ReLU-like responses. These findings reveal previously unexplored nonlinear dynamics in dual-pumped Kerr resonators and establish new pathways towards compact, all-optical neuromorphic circuits.