On critical normal sections for two-dimensional immersions in R^n and a Riemann-Hilbert problem

TL;DR

This paper investigates critical normal sections of two-dimensional immersions in R^4, defining torsion coefficients and a total torsion functional, and provides estimates relating torsion to the normal bundle's curvature.

Contribution

It introduces a new functional for total torsion of normal sections and characterizes critical points, linking torsion coefficients to the normal bundle's curvature.

Findings

Critical normal sections satisfy specific torsion conditions.

A global estimate relates torsion coefficients to the normal bundle's curvature.

The work connects geometric properties of immersions with variational principles.

Abstract

For orthonormal normal sections of two-dimensional immersions in R^4 we define torsion coefficients and a functional for the total torsion. We discuss normal sections which are critical for this functional. In particular, a global estimate for the torsion coefficients of a critical normal section in terms of the curvature of the normal bundle is provided.