Indices of Some Meromorphic Functions of Degree $3$ on Tori

Abstract

The index of a meromorphic function g on a compact Riemann surface is an invariant of g, which is defined as the number of negative eigenvalues of the differential operator L:=—Δ—| dG |2, where Δ is the Laplacian with respect to a conformal metric ds2 on the Riemann surface and G:M→S2 is the holomorphic map corresponding to g. We consider the meromorphic function w on the Riemann surface Ma={(z,w)∈ℂ^2|w2=z(z−a)(z+1a)}(a≥1) homeomorphic to a torus. We find a0>1 and determine the index of tw for all a in the range 1≤a≤a0 and all t>0.