A global invariant of conformal mappings in space

Abstract

This paper shows that the total integral of the square of the mean curvature for a compact orientable surface in E 3 {E^3} is an invariant of a conformal space mapping. This result is then used to answer a problem raised by T. Willmore and B.-Y. Chen concerning embeddings of compact orientable surfaces, and in particular tori, for which this integral is a minimum.