Co-rotating vortices on a catenoid rotate rigidly at fixed latitude with angular velocity proportional to the Gaussian curvature gradient, exhibiting linear instability and secular azimuthal drift in generic cases.
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Dissipative vortex binaries on a torus show outward spiraling for same-sign pairs, geometry-induced angular drift for dipoles, and a finite-time nonlinear chirp with frequency scaling as the square of itself for unequal opposite-sign pairs.