Z. 3. 4) where o(x,y)/ o(x,ji) is the determinant of the Jacobian. 4) has the simple form I(z) _ 2 = I(z(z)) Idz di l . 2.

1J = - = Lw frJ. 4. Proposition. Let rJ. E L 2(M) be of class C 1. Then rJ. is exact (respectively, co-exact) if and only if (a,f3) = 0 for all co-closed (closed) smooth differentials {3 of compact support. If rJ. is C 1 and exact, then rJ. = df with f of class C 2 • If {3 is co-closed, smooth, with support in D (with Cl D compact), then PROOF. ,{3) = ffD df /\ *13 = ffD [d(f*7J) = LD f *13 = o. 2) that Je IX = 0 for all simple closed curves. 1). The assertion for the co-exact differentials follows from the part of the proposition already established.

7 we see that in order to construct harmonic differentials, we must construct closed differentials that are not exact. Let c be a simple closed curve on the Riemann surface M that does not separate (that is, M \ c is connected). 4. Ha rmon ic Differentials c* (a dual curve to c) by starting on the right( + ) side of c and ending up on the other ( = left ( - )) side of c. 7). The curve c* will intersect c in exactly one point P. 3, we see that S. IJc = lim f(Q) - Q_P _ C lim f(Q) = 1 Q_ P + (here, of course, limQ _ p+ means you approach P through Q + ).