  We introduce a family of variational functionals for spinor fields on a
compact Riemann surface $M$ that can be used to find close-to-conformal
immersions of $M$ into $\mathbb{R}^3$ in a prescribed regular homotopy class.
Numerical experiments indicate that, by taking suitable limits, minimization of
these functionals can also yield piecewise smooth isometric immersions of a
prescribed Riemannian metric on $M$.
