On Hodge decomposition and conformal variational problems

TL;DR

This paper uses Hodge decomposition to identify the orthogonal complement of conformal vector fields on flat 2-manifolds, deriving differential equations for related variational problems and reviewing known vector field decompositions.

Contribution

It provides a new identification of the orthogonal complement of conformal vector fields and derives associated differential equations, expanding understanding of variational problems on 2-manifolds.

Findings

Orthogonal complement of conformal vector fields identified

Differential equations for variational problems derived

Review of vector field decompositions included

Abstract

The main result is the identification of the orthogonal complement of the subalgebra of conformal vector field inside the algebra of all vector fields of a compact flat 2-manifold. As a fundamental tool, the complete Hodge decomposition for manifold with boundary is used. The identification allows the derivation of governing differential equations for variational problems on the space of conformal vector fields. Several examples are given. In addition, the paper also gives a review, in full detail, of already known vector field decompositions involving subalgebras of volume preserving and symplectic vector fields.