Harmonic functions | Differential forms | Riemann surfaces

Differential forms on a Riemann surface

In mathematics, differential forms on a Riemann surface are an important special case of the general theory of differential forms on smooth manifolds, distinguished by the fact that the conformal structure on the Riemann surface intrinsically defines a Hodge star operator on 1-forms (or differentials) without specifying a Riemannian metric. This allows the use of Hilbert space techniques for studying function theory on the Riemann surface and in particular for the construction of harmonic and holomorphic differentials with prescribed singularities. These methods were first used by in his variational approach to the Dirichlet principle, making rigorous the arguments proposed by Riemann. Later found a direct approach using his method of orthogonal projection, a precursor of the modern theory of elliptic differential operators and Sobolev spaces. These techniques were originally applied to prove the uniformization theorem and its generalization to planar Riemann surfaces. Later they supplied the analytic foundations for the harmonic integrals of . This article covers general results on differential forms on a Riemann surface that do not rely on any choice of Riemannian structure. (Wikipedia).

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Michael Atiyah | Complex differential form | Dirichlet's principle | Tubular neighborhood | Functional analysis | Uniformization theorem | Homotopy | Product rule | Hodge star operator | Indicator function | Lattice (discrete subgroup) | Fourier series | Fourier analysis | Lattice (order) | Sobolev space | Hodge theory | Inverse function theorem | Richard Courant | Symmetric space | Beltrami equation | Planar Riemann surface | De Rham cohomology | Riemann surface | Binomial theorem | Mathematics | Cauchy–Riemann equations | Distribution (mathematics) | Intersection number | Orthonormal basis | Riemannian manifold | Differential forms on a Riemann surface | Partition of unity | Harmonic function | Compact operator | Hilbert space | Hermann Weyl | Bernhard Riemann | Moduli space | Fundamental solution | Trigonometric polynomial | Harmonic analysis | Green's theorem