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By Wehrfritz B.A.F.

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Recall that Ai (λ) = λ Ai Bi (λ) = λ Bi for all smooth one forms λ on S . It follows from our hypothesis that g ω = i=1 g η = k=1 g Ai (ω)αi + Ak (η)αk + j=1 g Bj (ω)βj + dφ B (η)β + dψ =1 Here, φ and ψ are smooth functions on S . 5, g S ω∧η = η− ω i=1 Ai Bi ω η Bi Ai ω ∧ dψ + dφ ∧ η − dφ ∧ dψ + S By Stoke’s theorem, S ω ∧ dψ + dφ ∧ η − dφ ∧ dψ = S d − ψω + φη + ψdφ = ∂S = −ψω + φη + ψdφ φη ∂S since dφ = ω on a neighborhood of ∂S . If f is any smooth function on a neighborhood U of ∂S with df = ω U then there are constants c1 , · · · , cm with f for i = 1, · · · , m .

7 Let X be a parabolic Riemann surface. A harmonic function f on X satisfying df 2 = X df ∧ ∗df < ∞ is constant. 3B], or the appendix to this section. 8 Let h be an exhaustion function with finite charge on the marked Riemann surface (X; A1 , B1 , · · ·) . Then, X is parabolic and (X; A1 , B1 , · · ·) has a unique, normalized basis ωk , k ≥ 1 , of square integrable holomorphic one forms dual to A k , k ≥ 1 . Furthermore, the period map 1 (σ, ω) ∈ H1 (X, C) × HHK (X) −→ σ ω ∈ C 1 (X) . is a well-defined, nondegenerate pairing of H1 (X, C) and HHK The next topic is the canonical map.

It follows that η = 0 . We now generalize the fact that the canonical map for a compact Riemann surface is an embedding if and only if the surface is not hyperelliptic. 16 A Riemann surface is hyperelliptic if there is a finite subset I of IP1 (C) , a discrete subset S of IP1 (C) \ I and a proper holomorphic map τ from X to IP1 (C) \ I of degree two that ramifies precisely over the points of S . The map τ is called the hyperelliptic projection for X . The holomorphic map ıX from X to X characterized by τ ıX (x) x, ı(x) = τ (x) = (x, x) (x, ıX (x) = x) , x ∈ τ −1 (S) ,x∈ / τ −1 (S) is called the hyperelliptic involution.

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