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It is shown that time-independent solutions to the (2+1)-dimensional non-linear O(3) sigma model may be placed in correspondence with surfaces ofconstant mean curvature in three-dimensional Euclidean space. The toolsrequired to establish this correspondence are provided by the classicaldifferential geometry of surfaces. A constant-mean-curvature surface induces asolution to the O(3) model through the identification of the Gauss map, ornormal vector, of the surface with the field vector of the sigma model. Someexplicit solutions, including the solitons and antisolitons discovered byBelavin and Polyakov, and a more general solution due to Purkait and Ray, areconsidered and the surfaces giving rise to them are found explicitly. It isseen, for example, that the Belavin-Polyakov solutions are induced by the Gaussmaps of surfaces which are conformal to their spherical images, i.e. spheresand minimal surfaces, and that the Purkait-Ray solution corresponds to thefamily of constant-mean-curvature helicoids first studied by do Carmo andDajczer in 1982. A generalisation of this method to include time-dependence mayshed new light on the role of the Hopf invariant in this model.Comment: 35 pages, LaTe

TIME-INDEPENDENT SOLUTIONS TO THE TWO-DIMENSIONAL NONLINEAR O(3) SIGMA MODEL AND SURFACES OF CONSTANT MEAN CURVATURE