International Congress of Mathematicians

Value Distribution and Potential Theory 687Theorem 10 [8] Let Di,...,D q be divisors and f a curve satisfying all the hypothesesof Theorem 7. Suppose in addition that f has finite lower order and thatequality holds in the defect relation (3.4). Then(i) 2X is an integer, and X > 1,(ii) T(r,f) = r x £(r), where £(r) is a slowly varying function in the sense of Karamata:£(cr)/£(r) —¥ 1, r —t oo uniformly with respect to c £ [1,2],(iii) All defects are rational: ö(Dj,f) = Pj/X, where pj are integers whose sum is2mX.When m = n = 1, this result was conjectured by F. Nevanlinna [23]. Afterlong efforts, mainly by A. Pfluger, A. Edrei, W. Fuchs and A. Weitsman, D. Drasinfinally proved F. Nevanlinna's conjecture in [4]. The potential-theoretic methodpresented here permitted to give a simpler proof of Drasin's theorem, and then togeneralize the result to arbitrary dimension, as well as to obtain a stronger resultin dimension 1 which is discussed in the next section. The proof of Theorem 10, isbased on the following result about subharmonic functions:Theorem 11 Suppose that v,vi,...,v q , q> 2m + 1 are subharmonic functions inthe plane, which satisfy (3.2), and in addition v(z) < \z\ x , z £ G, and v(0) = 0.Then the functionih = 2_, v j — 2TOWj=iis subharmonic. If h is harmonic, then 2X is an integer andwhere c > 0 and a is a real constant.v(re %t ) = c|r| A |cosA(r. — a)\,4. Functions with small ramificationWe recall the definition of the ramification term in Nevanlinna theory. Supposethat the image /(C) of a holomorphic curve / : C —¥ P" is not contained inany hyperplane. This means that / 0 ,...,/„ in the homogeneous representationof / are linearly independent. Let rii (r, /) be the number of zeros in the disc{z : \z\ < r} of the Wronski determinant W(fo, • • •, f n ), and Ni(r, /) the averagedcounting function of these zeros as in (3.3). If n = 1, then m counts the number ofcritical points of /. The Second Alain Theorem of Cartan [18] says that for everyholomorphic curve / whose image does not belong to a hyperplane, and every finiteset of hyperplanes {ai,... ,a q } in general position, we have(q^n-l + o(lj)T(r,f) + Ni(r,f)