## Riemann Surfaces of Infinite Genus |

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0reg Abel-Jacobi map analytic assume Banach space biholomorphic bounded charge canonical homology basis choose codimension compact subset complement component const constant construction contains converges coordinate Corollary define Definition 20.2 derivative det2 diffeomorphism differential dividing cycle divisor equation exhaustion function finite charge finite number follows function h function with finite Furthermore geometric hypotheses H(tj handle harmonic heat curve Hilbert space holomorphic function holomorphic map holomorphic one forms hyperelliptic hypothesis implies infinite genus intersection Lemma marked Riemann surface meromorphic function Morse function Nb(k neighbourhood nontrivial nonzero norm obeys Observe open subset parabolic Riemann surface path Pb(k period matrix Proof of Theorem Proposition 6.16 resp satisfying sequence si(j smooth solution square integrable sufficiently large Suppose Theorem 6.4 theta function theta series Torelli theorem vector Xcom zero