# Finding Conformal and Isometric Immersions of Surfaces

## TL;DR

This paper introduces variational functionals for spinor fields to find conformal and isometric immersions of surfaces into three-dimensional space, with numerical methods supporting the approach.

## Contribution

It proposes a novel family of variational functionals for spinor fields to compute conformal and isometric immersions of surfaces.

## Key findings

- Functional minimization yields close-to-conformal immersions
- Numerical experiments produce piecewise smooth isometric immersions
- Method can handle prescribed Riemannian metrics

## Abstract

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 $M$.