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Topological spin excitations induced by an external magnetic field coupled to a surface with rotational symmetry

We study the Heisenberg model in an external magnetic field on curved surfaces with rotational symmetry. The Euler-Lagrange static equations, derived from the Hamiltonian lead to the inhomogeneous double sine-Gordon equation (DSG). However, if the magnetic field is coupled with the metric elements of the surface, and consequently, its curvature, the homogeneous DSG appears and a $2π$-soliton is obtained as a solution for this model. In order to obey the self-dual equations, surface deformations are predicted at the sector where the spins point in the opposite direction to the magnetic field. The model was used to particularize the characteristic lenght of the 2$π$-soliton for three specific rotationally symmetric surfaces: the cylinder, the catenoid and the hyperboloid. Fractional 2$π$-solitons must appear on finite surfaces, as the sphere, torus and barrels, for example.