A THEOREM OF DIFFERENTIAL MAPPINGS OF RIEMANN SURFACES

In this paper, we have extended S.S. Chern's second basic theorem about holomor- phic mapping between two Riemann surfaces to more general case, and also obtained two similar results.

Let M be a compact Riemann surface, G be a Hermitian metric of M which has constant Gauss curvature K. Let ft be the volume element of G.For every a E M, Chern has proved that there exsits a real function U which is C on M-{a}, and satisfies: where ha Ua i(-0) If z is a local coordinate function on the neighbour U of a, such that z(a) 0, then Ua(z)+ log Izl is C on U.In [5], Chern proves the following theorems.
THEOREM A. Let D be a compact differentiable, orientable domain bounded by a sectiuonally smooth curve OD, f D M is a differential mapping, if a E M such that, f-{a} fq OD , and f-{a} is a finite set of points, then we have: THEOREM B, Let D be compact Riemann surface with smooth boundary 0D, f D M is a holomorphic mapping, then x(D)o where x(D) and x(M) are Euler's characteristics of D and M respectively, n(D) is the station- ary index of f in D.
For the :ase of holomorphic mappings, S.S.Chern gave the integral form of theorem A and theorem B, and also proved the ralation inequlity of deficient values.In this paper, we replace f by differential mapping, and also get similar results.We have the following main result.
THEOREM 1.Let D be compact Riemann surface with smooth boundary OD, if f D M is a differential mapping, and if the critical points of f are all isolated points, f is orientation-preserving except critical points.Then we also have equality (3).
By using local coordinate z x + yi, we have G gdzd2 g(dx + dye), and ft gdz A d 9dx A dy, where 9 is a positive function which belongs to C.
Lemma 1 (Gauss-Bonnet formula).If A is a compact subset of M with smooth boundary 0A, let K Kgds be the curvature form of 0A about G, where h'g is the curvature of 0A then especially, x(M) fM Now we define stationary index nl(D) of differential mapping of f D M as the following: We suppose that crl,c,...,a, are the critical points of f in D-OD i.e. dr(%) O,j 1,2,..., n).Because f is orientation-perserving except the critical points), then the metric of G on M can induce Hermitian metric f*G on D {al,... ,a,,}, f: D {a,,..., a,,} M is local isometry mapping, so f*(Kfl)is equal to the product of Gauss curvature of and volume element of fiG.We suppose that z, is the local coordinate function in the neigbohour of , such that z,(o,) 0, W, {Iz] < }, W U;'=, W. We use K denote the geodesio curvature form of OW about fiG.
Now we define stationary index I, of f at cq as the following: fo K-l, io, w.
We notice that K is constant, then apply Gauss-Bonnet formula, we have:

So we have
We are done. In Let V be a open Riemann surface.Suppose that V has an infinite harmonic exhaustion function v [13].We also suppose that f V M is a differential mapping, and all critical points are isolated points, f is orientation-preserving (except critical points), if is one-form on V, let *W be the conjugate one-form.
We let U[r] {pit(p) < r}, if r is not the critical values of r, then U[r] is compact subset in U with smooth boundary.Let n(r,a) n(V[r],a), v(r) v(U[r]), (r) (U[r]), n(r) n(U[r]).
For f, we use theorem l, we conclude Jo K + n,(r) )4(M)v(r), (4) () ,tl whcre K is geodesio curvature form of OV[r] about f'G.We can introduce function h such that, f*fl hdr A ,dr on V U[r(r)], because f is orientation-persveing, so f*fl and V have the same orientation, clearly, dr A ,dr and V have the same orientation, so h is nonnegative function, and K 71dClogh, then we have fov[rlK 7l fov[d dClog(h).According to [13], we can use special coordinate function a r + ip, so dClog(h) O---d log(h) + O l g ( h ) d p o r and 1L dClog(h )_ 1 0log(h) 0 1L log(h)*dr).By using the method which we deal with holomorphic functions, we can introduce the following functions: E(r) rio x(t)dt, Nl(r) rio n(t)dt, and T(r) rio v(t)dt, where r > r0 >_ r(r).