Morse's index formula in VMO for compact manifolds with boundary

TL;DR

This paper develops a topological index for VMO vector fields on compact manifolds with boundary, extending Morse's index formula and characterizing boundary data extendability, with applications to nematic liquid crystal modeling.

Contribution

It introduces a new topological invariant for VMO vector fields and establishes an analogue of Morse's formula for manifolds with boundary.

Findings

Constructed a topological index for VMO vector fields.

Established Morse's index formula analogue in VMO setting.

Characterized boundary data extendability for nowhere vanishing VMO vector fields.

Abstract

In this paper, we study Vanishing Mean Oscillation vector fields on a compact manifold with boundary. Inspired by the work of Brezis and Niremberg, we construct a topological invariant - the index - for such fields, and establish the analogue of Morse's formula. As a consequence, we characterize the set of boundary data which can be extended to nowhere vanishing VMO vector fields. Finally, we show briefly how these ideas can be applied to (unoriented) line fields with VMO regularity, thus providing a reasonable framework for modelling a surface coated with a thin film of nematic liquid crystals.