Riemann bilinear relations on minimal surfaces

Abstract

From a physical point of view, minimal surfaces in R are objects submitted to a balanced force system, consisting in the forces associated to non zero onedimensional homology classes in the surface. Several geometric properties can be studied in terms of those forces, as embeddedness, symmetries and deformations, leading up to uniqueness and non existence results, see [11–13]. More precisely, each closed curve γ in a minimal surface S ⊂ R carries a force that expresses the stress produced by an unit conormal vector field η along γ on the whole surface. The action of this conormal field provides a tendency of translation, or linear momentum F , and another one of rotation around an axis, or angular momentum M . The first one is given by the force vector ∫