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Spectrum of spin eigenmodes localized on a ferromagnetic skyrmion pinned by ageometrical defect (bump) of magnetic films is studied theoretically. By meansof direct numerical solution of the corresponding eigenvalue problem and finiteelement micromagnetic simulations we demonstrate, that the curvature can inducelocalized modes with higher azimuthal and radial quantum numbers, which areabsent for planar skyrmions (for the same parameters). The eigenfrequencies ofall modes, except the breathing and gyromodes decreases with increasingcurvature. Due to the translational symmetry break, the zero translational modeof the skyrmion gains a finite frequency and forms the gyromode, whichdescribes the uniform rotation of skyrmions around the equilibrium position. Inorder to treat the gyromotion analytically we developed a Thiele-likecollective variable approach. We show that N\'{e}el skyrmions in curvilinearfilms experience a driving force originating from the gradient of the meancurvature. The gyrofrequency of the pinned skyrmion is proportional to thesecond derivative of the mean curvature at the point of equilibrium.

The effect of curvature on the eigenexcitations of magnetic skyrmions