Finding Conformal and Isometric Immersions of Surfaces

Albert Chern
TU Berlin
Franz Pedit
UMass Amherst
Ulrich Pinkall
TU Berlin
Peter Schröder
Caltech

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 \(\Bbb R^3\) 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 \(M\).

Authors supported by SFB Transregio 109 "Discretization in Geometry and Dynamics" at Technical University Berlin. Third author partially supported by an RTF grant from the University of Massachusetts Amherst. Fifth author partially supported by the Einstein Foundation. Software support for images provided by SideFX. We thank Stefan Sechelmann for the abstract hyperbolic triangulated surface used in Figure 1.