Differential forms on a Riemann surface

Hodge star on 1-forms

On a Riemann surface the Hodge star is defined on 1-forms by the local formula

⋆ ( p d x + q d y ) = − q d x + p d y .

It is well-defined because it is invariant under holomorphic changes of coordinate.

Indeed, if z = x + iy is holomorphic as a function of w = u + iv, then by the Cauchy–Riemann equations x_u = y_v and y_u = – x_v. In the new coordinates

p d x + q d y = ( p x u + q y u ) d u + ( p x v + q y v ) d v = p 1 d u + q 1 d v ,

so that

− q 1 d u + p 1 d v = − ( p x v + q y v ) d u + ( p x u + q y u ) d v = − q ( x u d u + x v d v ) + p ( y u d u + y v d v ) = − q d x + p d y ,

proving the claimed invariance.

Note that for 1-forms ω₁ = p₁ dx + q₁ dy and ω₂ = p₂ dx + q₂ dy

ω 1 ∧ ⋆ ω 2 = ( p 1 p 2 + q 1 q 2 ) d x ∧ d y = ω 2 ∧ ⋆ ω 1 .

In particular if ω = p dx + q dy then

ω ∧ ⋆ ω = ( p 2 + q 2 ) d x ∧ d y .

Note that in standard coordinates

⋆ d z = − i d z , ⋆ d z ¯ = i d z ¯ .

Recalling that

∂ ∂ z = 1 2 ( ∂ ∂ x − i ∂ ∂ y ) , ∂ ∂ z ¯ = 1 2 ( ∂ ∂ x + i ∂ ∂ y ) ,

so that

d f = f z d z + f z ¯ d z ¯ = ∂ f + ∂ ¯ f .

The decomposition d = ∂ + ∂ ¯ is independent of the choice of local coordinate. The 1-forms with only a d z component are called (1,0) forms; those with only a d z ¯ component are called (0,1) forms. The operators ∂ and ∂ ¯ are called the Dolbeault operators.

It follows that

⋆ d f = − i ∂ f + i ∂ ¯ f .

The Dolbeault operators can similarly be defined on 1-forms and as zero on 2-forms. They have the properties

d = ∂ + ∂ ¯ ∂ 2 = ∂ ¯ 2 = ∂ ∂ ¯ + ∂ ¯ ∂ = 0.

Poincaré lemma

On a Riemann surface the Poincaré lemma states that every closed 1-form or 2-form is locally exact. Thus if ω is a smooth 1-form with dω = 0 then in some open neighbourhood of a given point there is a smooth function f such that ω = df in that neighbourhood; and for any smooth 2-form Ω there is a smooth 1-form ω defined in some open neighbourhood of a given point such that Ω = dω in that neighbourhood.

If ω = p dx + q dy is a closed 1-form on (a,b) × (c,d), then p_y = q_x. If ω = df then p = f_x and q = f_y. Set

g ( x , y ) = ∫ a x p ( t , y ) d t ,

so that g_x = p. Then h = f − g must satisfy h_x = 0 and h_y = q − g_y. The right hand side here is independent of x since its partial derivative with respect to x is 0. So

h ( x , y ) = ∫ c y q ( a , s ) d s − g ( a , y ) = ∫ c y q ( a , s ) d s ,

and hence

f ( x , y ) = ∫ a x p ( t , y ) d t + ∫ c y q ( a , s ) d s .

Similarly if Ω = r dx ∧ dy then Ω = d(f dx + g dy) with g_x − f_y = r. Thus a solution is given by f = 0 and

g ( x , y ) = ∫ a x r ( t , y ) d t .

Comment on differential forms with compact support. Note that if ω has compact support, so vanishes outside some smaller rectangle (a₁,b₁) × (c₁,d₁) with a < a₁ < b₁ <b and c < c₁ < d₁ < d, then the same is true for the solution f(x,y). So the Poincaré lemma for 1-forms holds with this additional conditions of compact support.

A similar statement is true or 2-forms; but, since there is some choices for the solution, a little more care has to be taken in making those choices.

In fact if Ω has compact support on (a,b) × (c,d) and if furthermore ∬ Ω = 0, then Ω = dω with ω a 1-form of compact support on (a,b) × (c,d). Indeed, Ω must have support in some smaller rectangle (a₁,b₁) × (c₁,d₁) with a < a₁ < b₁ <b and c < c₁ < d₁ < d. So r(x, y) vanishes for x ≤ a₁ or x ≥ b₁ and for y ≤ c₁ or y ≥ d₁. Let h(y) be a smooth function supported in (c₁,d₁) with ∫d
c h(t) dt = 1. Set k(x) = ∫d
c r(x,y) dy: it is a smooth function supported in (a₁,b₁). Hence R(x,y) = r(x,y) − k(x)h(y) is smooth and supported in (a₁,b₁) × (c₁,d₁). It now satisfies ∫d
c R(x,y) dy ≡ 0. Finally set

P ( x , y ) = ∫ c y R ( x , y ) d y , Q ( x , y ) = h ( y ) ∫ a x k ( s ) d s .

Both P and Q are smooth aand supported in (a₁,b₁) × (c₁,d₁) with P_y =R and Q_x(x,y)= k(x)h(y). Hence ω = −P dx + Q dy is a smooth 1-form supported in (a₁,b₁) × (c₁,d₁) with

d ω = ( Q x + P y ) d x ∧ d y = r d x ∧ d y = Ω .

Integration of 2-forms

If Ω is a continuous 2-form of compact support on a Riemann surface X, its support K can be covered by finitely many coordinate charts U_i and there is a partition of unity χ_i of smooth non-negative functions with compact support such that ∑ χ_i = 1 on a neighbourhood of K. Then the integral of Ω is defined by

∫ X Ω = ∑ ∫ X χ i Ω = ∑ ∫ U i χ i Ω ,

where the integral over U_i has its usual definition in local coordinates. The integral is independent of the choices here.

If Ω has the local representation f(x,y) dx ∧ dy, then |Ω| is the density |f(x,y)| dx ∧ dy, which is well defined and satisfies |∫_X Ω| ≤ ∫_X |Ω|. If Ω is a non-negative continuous density, not necessarily of compact support, its integral is defined by

∫ X Ω = sup 0 ≤ ψ ≤ 1 , ψ ∈ C c ( X ) ∫ X ψ Ω .

If Ω is any continuous 2-form it is integrable if ∫_X |Ω| < ∞. In this case, if ∫_X |Ω| = lim ∫_X ψ_n |Ω|, then ∫_X Ω can be defined as lim ∫_X ψ_n Ω. The integrable continuous 2-forms form a complex normed space with norm ||Ω||₁ = ∫_X |Ω|.

Integration of 1-forms along paths

If ω is a 1-form on a Riemann surface X and γ(t) for a ≤ t ≤ b is a smooth path in X, then the mapping γ induces a 1-form γ∗ω on [a,b]. The integral of ω along γ is defined by

∫ γ ω = ∫ a b γ ∗ ω .

This definition extends to piecewise smooth paths γ by dividing the path up into the finitely many segments on which it is smooth. In local coordinates if ω = p dx + q dy and γ(t) = (x(t),y(t)) then

γ ∗ ω = p ( γ ( t ) ) x ˙ ( t ) d t + q ( γ ( t ) ) y ˙ ( t ) d t ,

so that

∫ γ ω = ∫ a b p ( γ ( t ) ) x ˙ ( t ) d t + q ( γ ( t ) ) y ˙ ( t ) d t .

Note that if the 1-form ω is exact on some connected open set U, so that ω = df for some smooth function f on U (unique up to a constant), and γ(t), a ≤ t ≤ b, is a smooth path in U, then

∫ γ ω = ∫ a b d ( f ( γ ( t ) ) = f ( γ ( b ) ) − f ( γ ( a ) ) .

This depends only on the difference of the values of f at the endpoints of the curve, so is independent of the choice of f. By the Poincaré lemma, every closed 1-form is locally exact, so this allows ∫_γ ω to be computed as a sum of differences of this kind and for the integral of closed 1-forms to be extended to continuous paths:

Monodromy theorem. If ω is a closed 1-form, the integral ∫_γ ω can be extended to any continuous path γ(t), a ≤ t ≤ b so that it is invariant under any homotopy of paths keeping the end points fixed.

The same argument shows that a homotopy between closed continuous loops does not change their integrals over closed 1-forms. Since ∫_γ df = f(γ(b)) − f(γ(a)), the integral of an exact form over a closed loop vanishes. Conversely if the integral of a closed 1-form ω over any closed loop vanishes, then the 1-form must be exact.

A closed 1-form is exact if and only if its integral around any piecewise smooth or continuous Jordan curve vanishes.

The above argument also shows that given a continuous Jordan curve γ(t), there is a finite set of simple smooth Jordan curves γ_i(t) with nowhere zero derivatives such that

∫ γ ω = ∑ i ∫ γ i ω

for any closed 1-form ω. Thus to check exactness of a closed form it suffices to show that the vanishing of the integral around any regular closed curve, i.e. a simple smooth Jordan curve with nowhere vanishing derivative.

Green–Stokes formula

If U is a bounded region in the complex plane with boundary consisting of piecewise smooth curves and ω is a 1-form defined on a neighbourhood of the closure of U, then the Green–Stokes formula states that

∫ ∂ U ω = ∫ U d ω .

In particular if ω is a 1-form of compact support on C then

∫ C d ω = 0 ,

since the formula may be applied to a large disk containing the support of ω.

Similar formulas hold on a Riemann surface X and can be deduced from the classical formulas using partitions of unity. Thus if U ⊂ X is a connected region with compact closure and piecewise smooth boundary ∂U and ω is a 1-form defined on a neighbourhood of the closure of U, then the Green–Stokes formula states that

∫ ∂ U ω = ∫ U d ω .

Moreover, if ω is a 1-form of compact support on X then

∫ X d ω = 0.

To prove the second formula take a partition of unity ψ_i supported in coordinate charts covering the support of ω. Then ∫_X dω = ∑ ∫_X d(ψ_i ω) = 0, by the planar result. Similarly to prove the first formula it suffices to show that

∫ ∂ U ψ ω = ∫ U d ( ψ ω )

when ψ is a smooth function compactly supported in some coordinate patch. If the coordinate patch avoids the boundary curves, both sides vanish by the second formula above. Otherwise it can be assumed that the coordinate patch is a disk, the boundary of which cuts the curve transversely at two points. The same will be true for a slightly smaller disk containing the support of ψ. Completing the curve to a Jordan curve by adding part of the boundary of the smaller disk, the formula reduces to the planar Green-Stokes formula.

The Green-Stokes formula implies an adjoint relation for the Laplacian on functions defined as Δf = −d∗df. This gives a 2-form, given in local coordinates by the formula

Δ f = ( − ∂ 2 f ∂ x 2 − ∂ 2 f ∂ y 2 ) d x ∧ d y .

Then if f and g are smooth and the closure of U is compact

∫ U f Δ g − g Δ f = ∫ ∂ U g ⋆ d f − f ⋆ d g .

Moreover, if f or g has compact support then

∫ X f Δ g = ∫ X g Δ f .

Duality between 1-forms and closed curves

Theorem. If γ is a continuous Jordan curve on a Riemann surface X, there is a smooth closed 1-form α of compact support such that ∫_γ ω = ∫_X ω ∧ α for any closed smooth 1-form ω on X.

Corollary. A closed smooth 1-form ω is exact if and only if ∫_X ω ∧ α = 0 for all smooth 1-forms α of compact support.

Holomorphic and harmonic 1-forms

A holomorphic 1-form ω is one that in local coordinates is given by an expression f(z) dz with f holomorphic. Since d g = ∂ z g d z + ∂ z ¯ g d z ¯ , it follows that dω = 0, so any holomorphic 1-form is closed. Moreover, since ∗dz = i dz, ω must satisfy ∗ω = iω. These two conditions characterize holomorphic 1-forms. For if ω is closed, locally it can be written as dg for some g, The condition ∗dg = i dg forces ∂ z ¯ g = 0 , so that g is holomorphic and dg = g '(z) dz, so that ω is holomorphic.

Let ω = f dz be a holomorphic 1-form. Write ω = ω₁ + iω₂ with ω₁ and ω₂ real. Then dω₁ = 0 and dω₂ = 0; and since ∗ω = iω, ∗ω₁ = −ω₂. Hence d∗ω₁ = 0. This process can clearly be reversed, so that there is a one-one correspondence between holomorphic 1-forms and real 1-forms ω₁ satisfying dω₁ = 0 and d∗ω₁ = 0. Under this correspondence, ω₁ is the real part of ω while ω is given by ω= ω₁ + i∗ω₁. Such forms ω₁ are called harmonic 1-forms. By definition ω₁ is harmonic if and only if ∗ω₁ is harmonic.

Since holomorphic 1-forms locally have the form df with f a holomorphic function and since the real part of a holomorphic function is harmonic, harmonic 1-forms locally have the form dh with h a harmonic function. Conversely if ω₁ can be written in this way locally, d∗ω₁ = d∗dh = (h_xx + h_yy) dx∧dy so that h is harmonic.

Remark. The definition of harmonic functions and 1-forms is intrinsic and only relies on the underlying Riemann surface structure. If, however, a conformal metric is chosen on the Riemann surface (see below), the adjoint d* of d can be defined and the Hodge star operation extended to functions and 2-forms. The Hodge Laplacian can be defined on k-forms as ∆_k = dd* +d*d and then a function f or a 1-form ω is harmonic if and only if it is annihilated by the Hodge Laplacian, i.e. ∆₀f = 0 or ∆₁ω = 0. The metric structure, however, is not required for the application to the uniformization of simply connected or planar Riemann surfaces.

Sobolev spaces on T2

The theory of Sobolev spaces on T² can be found in Bers, John & Schechter (1979), an account which is followed in several later textbooks such as Warner (1983) and Griffiths & Harris (1994). It provides an analytic framework for studying function theory on the torus C/Z+i Z = R² / Z² using Fourier series, which are just eigenfunction expansions for the Laplacian –∂²/∂x² –∂²/∂y². The theory developed here essentially covers tori C / Λ where Λ is a lattice in C. Although there is a corresponding theory of Sobolev spaces on any compact Riemann surface, it is elementary in this case, because it reduces to harmonic analysis on the compact Abelian group T². Classical approaches to Weyl's lemma use harmonic analysis on the non-compact Abelian group C = R², i.e. the methods of Fourier analysis, in particular convolution operators and the fundamental solution of the Laplacian.

Let T² = {(e^ix,e^iy: x, y ∈ [0,2π)} = R²/Z² = C/Λ where Λ = Z + i Z. For λ = m + i n ≅ (m,n) in Λ, set e_λ (x,y) = e^{i(mx + ny)}. Furthermore, set D_x= -i∂/∂x and D_y = -i∂/∂y. For α = (p,q) set D^α =(D_x)^p (D_y)^q, a differential operator of total degree |α| = p + q. Thus D^αe_λ = λ^α e_λ, where λ^α =m^pn^q. The (e_λ) form an orthonormal basis in C(T²) for the inner product (f,g) = (2π)⁻²∬ f(x,y) g(x,y) dx dy, so that (∑ a_λ e_λ, ∑ b_μ e_μ) = ∑ a_λb_λ.

For f in C^∞(T'²) and k an integer, define the kth Sobolev norm by

∥ f ∥ ( k ) = ( ∑ | f ^ ( λ ) | 2 ( 1 + | λ | 2 ) k ) 1 / 2 .

The associated inner product

( f , g ) ( k ) = ∑ f ^ ( λ ) g ^ ( λ ) ¯ ( 1 + | λ | 2 ) k

makes C^∞(T²) into an inner product space. Let H_k(T²) be its Hilbert space completion. It can be described equivalently as the Hilbert space completion of the space of trigonometric polynomials—that is finite sums (∑ a_λ e_λ—with respect to the kth Sobolev norm, so that H_k(T²) = {∑ a_λ e_λ : ∑ |a_λ|²(1 + |λ|²)^k < ∞} with inner product

(∑ a_λ e_λ, ∑ b_μ e_μ)_(k) = ∑ a_λb_λ (1 + |λ|²)^k.

As explained below, the elements in the intersection H_∞(T²) = ∩ H_k(T²) are exactly the smooth functions on T²; elements in the union H_−∞(T²) = ∪ H_k(T²) are just distributions on T² (sometimes referred to as "periodic distributions" on R²).

The following is a (non-exhaustive) list of properties of the Sobolev spaces.

Differentiability and Sobolev spaces. C^k(T²) ⊂ H_k(T²) for k ≥ 0 since, using the binomial theorem to expand (1 + |λ|²)^k,

Differential operators. D^α H_k(T²) ⊂ H_k−|α|(T²) and D^α defines a bounded linear map from H_k(T²) to H_k−|α|(T²). The operator I + Δ defines a unitary map of H_k+2(T²) onto H_k(T²); in particular (I + Δ)^k defines a unitary map of H_k(T²) onto H_−k(T²) for k ≥ 0.

Duality. For k ≥ 0, the pairing sending f, g to (f,g) establishes a duality between H_k(T²) and H_−k(T²).

Multiplication operators. If h is a smooth function then multiplication by h defines a continuous operator on H_k(T²).

Sobolev spaces and differentiability (Sobolev's embedding theroem). For k ≥ 0, H_k+2(T²) ⊂ C^k(T²) and sup_|α|≤k |D^αf| ≤ C_k ⋅ ||f||_(k+2).

Smooth functions. C^∞(T²) = ∩ H_k(T²) consists of Fourier series ∑ a_λ e_λ such that for all k > 0, (1 + |λ|²)^k |a_λ| tends to 0 as |λ| tends to ∞, i.e. the Fourier coefficients a_λ are of "rapid decay".

Inclusion maps (Rellich's compactness theorem). If k > j, the space H_k(T²) is a subspace of H_j(T²) and the inclusion H_k(T²) → H_j(T²) is compact.

Elliptic regularity (Weyl's lemma). Suppose that f and u in H_−∞(T²) = ∪ H_k(T²) satisfy ∆u = f. Suppose also that ψ f is a smooth function for every smooth function ψ vanishing off a fixed open set U in T²; then the same is true for u. (Thus if f is smooth off U, so is u.)

Hodge decomposition on functions. H₀(T²) = ∆ H₂(T²) ⊕ ker ∆ and C^∞(T²) = ∆ C^∞(T²) ⊕ ker ∆.

Hodge theory on T². Let Ω^k(T²) be the space of smooth k-forms for 0 ≤ k ≤ 2. Thus Ω⁰(T²) = C^∞(T²), Ω¹(T²) = C^∞(T²) dx ⊕ C^∞(T²) dy and Ω²(T²) = C^∞(T²) dx ∧ dy. The Hodge star operation is defined on 1-forms by ∗(p dx + q dy) = −q dx + p dy. This definition is extended to 0-forms and 2-forms by *f = f dx ∧ dy and *(g dx ∧ dy) = g. Thus ** = (−1)^k on k-forms. There is a natural complex inner product on Ω^k(T²) defined by

Define δ = −∗d∗. Thus δ takes Ω^k(T²) to Ω^k−1(T²), annihilating functions; it is the adjoint of d for the above inner products, so that δ = d*. Indeed by the Green-Stokes formula The operators d and δ = d* satisfy d² = 0 and δ² = 0. The Hodge Laplacian on k-forms is defined by ∆_k = (d + d*)² = dd* + d*d. From the definition ∆₀ f = ∆f. Moreover ∆₁(p dx+ q dy) =(∆p)dx + (∆q)dy and ∆₂(f dx∧dy) = (∆f)dx∧dy. This allows the Hodge decomposition to be generalised to include 1-forms and 2-forms:

Hodge theorem. Ω^k(T²) = ker d ∩ ker d∗ ⊕ im d ⊕ im ∗d = ker d ∩ ker d* ⊕ im d ⊕ im d*. In the Hilbert space completion of Ω^k(T²) the orthogonal complement of im d ⊕ im ∗d is ker d ∩ ker d∗, the finite-dimensional space of harmonic k-forms, i.e. the constant k-forms. In particular in Ω^k(T²) , ker d / im d = ker d ∩ ker d*, the space of harmonic k-forms. Thus the de Rham cohomology of T² is given by harmonic (i.e. constant) k-forms.

Hilbert space of 1-forms

In the case of the compact Riemann surface C / Λ, the theory of Sobolev spaces shows that the Hilbert space completion of smooth 1-forms can be decomposed as the sum of three pairwise orthogonal spaces, the closure of exact 1-forms df, the closure of coexact 1-forms ∗df and the harmonic 1-forms (the 2-dimensional space of constant 1-forms). The method of orthogonal projection of Weyl (1940) put Riemann's approach to the Dirichlet principle on sound footing by generalizing this composition to arbitrary Riemann surfaces.

If X is a Riemann surface Ω1
c(X) denote the space of continuous 1-forms with compact support. It admits the complex inner product

( α , β ) = ∫ X α ∧ ⋆ β ¯ ,

for α and β in Ω1
c(X). Let H denote the Hilbert space completion of Ω1
c(X). Although H can be interpreted in terms of measurable functions, like Sobolev spaces on tori it can be studied directly using only elementary functional analytic techniques involving Hilbert spaces and bounded linear operators.

Let H₁ denote the closure of d C∞
c(X) and H₂ denote the closure of ∗d C∞
c(X). Since (df,∗dg) = ∫_X df ∧ dg = ∫_X d (f dg) = 0, these are orthogonal subspaces. Let H₀ denote the orthogonal complement (H₁ ⊕ H₂)^⊥ = H⊥
1 ∩ H⊥
2.

Theorem (Hodge−Weyl decomposition). H = H₀ ⊕ H₁ ⊕ H₂. The subspace H₀ consists of square integrable harmonic 1-forms on X, i.e. 1-forms ω such that dω = 0, d∗ω = 0 and ||ω||² = ∫_X ω ∧ ∗ω < ∞.

Every square integrable continuous 1-form lies in H.

If ω in H is such that ψ ⋅ ω is continuous for every ψ in C_c(X), then ω is a square integrable continuous 1-form.

Every square integrable harmonic 1-form ω lies in H₀.

Every element of H₀ is given by a square integrable harmonic 1-form.

From the formulas for the Dolbeault operators ∂ and ∂ ¯ , it follows that

d C c ∞ ( X ) ⊕ ∗ d C c ∞ ( X ) = ∂ C c ∞ ( X ) ⊕ ∂ ¯ C c ∞ ( X ) ,

where both sums are orthogonal. The two subspaces in the second sum correspond to the ±i eigenspaces of the Hodge ∗ operator. Denoting their closures by H₃ and H₄, it follows that H⊥
0 = H₃ ⊕ H₄ and that these subspaces are interchanged by complex conjugation. The smooth 1-forms in H₁, H₂, H₃ or H₄ have a simple description.

A smooth 1-form in H₁ has the form df for f smooth.

A smooth 1-form in H₂ has the form ∗df for f smooth.

A smooth 1-form in H₃ has the form ∂ f for f smooth.

A smooth 1-form in H₃ has the form ∂ ¯ f for f smooth.

The above characterisations have an immediate corollary:

A smooth 1-form α in H⊥
0 can be decomposed uniquely as α = da + ∗db = ∂ f + ∂ ¯ g, with a, b, f and g smooth and all the summands square integrable.

Combined with the previous Hodge–Weyl decomposition and the fact that an element of H₀ is automatically smooth, this immediately implies:

Theorem (smooth Hodge–Weyl decomposition). If α is a smooth square integrable 1-form then α can be written uniquely as α = ω + da + *db = ω + ∂ f + ∂ ¯ g with ω harmonic, square integrable and a, b, f, g smooth with square integrable differentials.

Holomorphic 1-forms with a double pole

The following result—reinterpreted in the next section in terms of harmonic functions and the Dirichlet principle—is the key tool for proving the uniformization theorem for simply connected, or more generally planar, Riemann surfaces.

Theorem. If X is a Riemann surface and P is a point on X with local coordinate z, there is a unique holomorphic differential 1-form ω with a double pole at P, so that the singular part of ω is z⁻²dz near P, and regular everywhere else, such that ω is square integrable on the complement of a neighbourhood of P and the real part of ω is exact on X {P}.

The double pole condition is invariant under holomorphic coordinate change z ↦ z + az² + ⋅ ⋅ ⋅. There is an analogous result for poles of order greater than 2 where the singular part of ω has the form z^–kdz with k > 2, although this condition is not invariant under holomorphic coordinate change.

Corollary of proof. If X is a Riemann surface and P is a point on X with local coordinate z, there is a unique real-valued 1-form δ which is harmonic on X {P} such that δ – Re z⁻²dz is harmonic near z = 0 (the point P) such that δ is square integrable on the complement of a neighbourhood of P. Moreover, if h is any real-valued smooth function on X with dh square integrable and h vanishing near P, then (δ,dh) = 0.

Dirichlet's principle on a Riemann surface

Theorem. If X is a Riemann surface and P is a point on X with local coordinate z, there is a unique real-valued harmonic function u on X {P} such that u(z) – Re z⁻¹ is harmonic near z = 0 (the point P) such that du is square integrable on the complement of a neighbourhood of P. Moreover, if h is any real-valued smooth function on X with dh square integrable and h vanishing near P, then (du,dh)=0.

This result can be interpreted in terms of Dirichlet's principle. Let D_R be a parametric disk |z| < R about P (the point z = 0) with R > 1. Let α = −d(ψz⁻¹), where 0 ≤ ψ ≤ 1 is a bump function supported in D = D₁, identically 1 near z = 0. Let α₁ = −χ_D(z) Re d(z⁻¹) where χ_D is the characteristic function of D. Let γ= Re α and γ₁ = Re α₁. Since χ_D can be approximated by bump functions in L², γ₁ − γ lies in the real Hilbert space of 1-forms Re H; similarly α₁ − α lies in H. Dirichlet's principle states that the distance function

F(ξ) = ||γ₁ − γ – ξ||

on Re H₁ is minimised by a smooth 1-form ξ₀ in Re H₁. In fact −du coincides with the minimising 1-form: γ + ξ₀ = -du.

This version of Dirichlet's principle is easy to deduce from the previous construction of du. By definition ξ₀ is the orthogonal projection of γ₁ – γ onto Re H₁ for the real inner product Re (η₁,η₂) on H, regarded as a real inner product space. It coincides with the real part of the orthogonal projection ω₁ of α₁ – α onto H₁ for the complex inner product on H. Since the Hodge star operator is a unitary map on H swapping H₁ and H₂, ω₂ = ∗ω₁ is the orthogonal projection of ∗(α₁ – α) onto H₂. On the other hand, ∗α₁ = −i α₁, since α is a (1,0) form. Hence

(α₁ – α) − i∗(α₁ – α) = ω₀ + ω₁ + ω₂,

with ω_k in H_k. But the left hand side equals – α + i∗α = −β, with β defined exactly as in the preceding section, so this coincides with the previous construction.

Further discussion of Dirichlet's principle on a Riemann surface can be found in Hurwitz & Courant (1929), Ahlfors (1947), Courant (1950), Schiffer & Spencer (1954), Pfluger (1957) and Ahlfors & Sario (1960).

Historical note. Weyl (1913) proved the existence of the harmonic function u by giving a direct proof of Dirichlet's principle. In Weyl (1940), he presented his method of orthogonal projection which has been adopted in the presentation above, following Springer (1957), but with the theory of Sobolev spaces on T² used to prove elliptic regularity without using measure theory. In the expository texts Weyl (1955) and Kodaira (2007), both authors avoid invoking results on measure theory: they follow Weyl's original approach for constructing harmonic functions with singularities via Dirichlet's principle. In Weyl's method of orthogonal projection, Lebesgue's theory of integration had been used to realise Hilbert spaces of 1-forms in terms of measurable 1-forms, although the 1-forms to be constructed were smooth or even analytic away from their singularity. In the preface to Weyl (1955), referring to the extension of his method of orthogonal projection to higher dimensions by Kodaira (1949), Weyl writes:

In Kodaira (2007), after giving a brief exposition of the method of orthogonal projection and making reference to Weyl's writings, Kodaira explains:

The methods of Hilbert spaces, L^p spaces and measure theory appear in the non-classical theory of Riemann surfaces (the study of moduli spaces of Riemann surfaces) through the Beltrami equation and Teichmüller theory.

Holomorphic 1-forms with two single poles

Theorem. Given a Riemann surface X and two distinct points A and B on X, there is a holomorphic 1-form on X with simple poles at the two points with non-zero residues having sum zero such that the 1-form is square integrable on the complement of any open neighbourhoods of the two points.

The proof is similar to the proof of the result on holomorphic 1-forms with a single double pole. The result is first proved when A and B are close and lie in a parametric disk. Indeed, once this is proved, a sum of 1-forms for a chain of sufficiently close points between A and B will provide the required 1-form, since the intermediate singular terms will cancel. To construct the 1-form for points corresponding to a and b in a parametric disk, the previous construction can be used starting with the 1-form

α = d ( ψ ( z ) log ⁡ z − a z − b ) ,

which locally has the form

d z z − a − d z z − b .

Poisson equation

Theorem (Poisson equation). If Ω is a smooth 2-form of compact support on a Riemann surface X, then Ω can be written as Ω = ∆f where f is a smooth function with df square integrable if and only if ∫_X Ω = 0.

In the case of the simply connected Riemann surfaces C, D and S= C ∪ ∞, the Riemann surfaces are symmetric spaces G / K for the groups G = R², SL(2,R) and SU(2). The methods of group representation theory imply the operator ∆ is G-invariant, so that its fundamental solution is given by right convolution by a function on K G / K. Thus in these cases Poisson's equation can be solved by an explicit integral formula. It is easy to verify that this explicit solution tends to 0 at ∞, so that in the case of these surfaces there is a solution f tending to 0 at ∞. Donaldson (2011) proves this directly for simply connected surfaces and uses it to deduce the uniformization theorem.