Get HARMONIC TWO-FORMS IN FOUR DIMENSIONS - Ams

Harmonic forms are a powerful tool in the study of the cohomology of real and complex manifolds and of cohomology spaces of discrete groups. The theory of harmonic forms yields fundamental cohomological properties of compact Kähler manifolds and, in particular, of projective algebraic varieties [1], [4], [5].

We can verify whether the given function is harmonic or not using the equation (∂2u/∂x2) + (∂2u/∂y2) = 0. If the given function satisfies this equation then it is a harmonic function.

Every harmonic form α on a closed Riemannian manifold is closed, meaning that dα = 0. As a result, there is a canonical mapping . The Hodge theorem states that. is an isomorphism of vector spaces. In other words, each real cohomology class on M has a unique harmonic representative.

Definition A differential form ω ∈ Ω n ( X ) on a Riemannian manifold ( X , g ) is called a harmonic form if it is in the kernel of the Laplace operator Δ g of X in that Δ ω = ( d + d † ) 2 ω = 0 .

The basic examples of harmonic functions are EXAMPLE 1.2 Here are several examples of harmonic functions in R2: 1. u(x, y) = x + y, o ⊂ R2 is any open set; 2. u(x, y) = x2 − y2, o ⊂ R2 is any open set; 3.