Conformal deformations of immersed discs in $R^3$ and elliptic boundary value problems

TL;DR

This paper explores the conditions under which Dirac-type boundary value problems for immersed surfaces in 3D Euclidean space are elliptic and self-adjoint, revealing spectral flow phenomena under boundary deformations.

Contribution

It characterizes ellipticity and self-adjointness of Dirac boundary problems in conformal surface geometry and analyzes spectral flow during boundary deformations.

Findings

Boundary value problems are elliptic and self-adjoint under specific conditions.

Spectral flow occurs during certain periodic boundary deformations.

Applications extend to computer graphics and geometric analysis.

Abstract

Boundary value problems for operators of Dirac type arise naturally in connection with the conformal geometry of surfaces immersed in Euclidean 3--space. Recently such boundary value problems have been successfully applied to a variety of problems from computer graphics. Here we investigate under which conditions these boundary value problems are elliptic and self--adjoint. We show that under certain periodic deformations of the boundary data our operators exhibit non-trivial spectral flow.