Two-dimensional Dirac operator and surface theory

TL;DR

This paper surveys the use of the two-dimensional Dirac operator and Weierstrass representations in surface theory, exploring their applications to the Willmore functional and spectral theory in geometric analysis.

Contribution

It provides a comprehensive overview of the connections between Dirac operators, surface representations, and spectral problems in differential geometry.

Findings

Connections between Dirac operators and surface representations clarified.

Applications to Willmore functional discussed.

Spectral theory of periodic Dirac operators explored.

Abstract

We give a survey on the Weierstrass representations of surfaces in three- and four-dimensional spaces, their applications to the theory of the Willmore functional and on related problems of spectral theory of the two-dimensional Dirac operator with periodic coefficients.