Computation of the index of some meromorphic functions of degree 3 on tori

概要

The index of a nonconstant 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 := −∆ − |dG| 2 , where ∆ is the Laplacian with respect to a conformal metric ds2 = λdζd¯ζ on the Riemann surface, defined by ∆ := 4 λ ∂ 2 ∂ζ∂ζ¯ using a local coordinate ζ, G: M → S 2 is the holomorphic map corresponding to the meromorphic function g and |dG| is the norm of the differential dG of G. The multiplicity of the eigenvalue 0 of L is called the nullity of g and denoted by Nul(g). The operator L depends on how to choose a conformal metric, but the index and the nullity do not depend on how to choose a conformal metric.

The index of a meromorphic function is closely related to the index (Morse index) of a complete minimal surface with finite total curvature. Huber [6] and Osserman [11] proved if the total curvature of a complete oriented minimal surface in R 3 is finite, this minimal surface is identified with a Riemann surface given by excluding finite points from a compact Riemann surface, and the Gauss map on this minimal surface is extended to a meromorphic function on the compact Riemann surface. Fischer-Colbrie[3] and GulliverLawson [4], [5] proved that for a complete oriented minimal surface in R 3 , the index is finite if and only if the total curvature is finite. This is a qualitative study of the index. Fischer-Colbrie proved when the total curvature a complete oriented minimal surface in R 3 is finite, the index coincides with the index of the extended Gauss map of this minimal surface. Tysk [12] proved the index of a complete oriented minimal surface in R 3 is bounded from above by some scalar multiple of the total curvature. This is the first quantitative study of the relationship between the index and the total curvature. Study of lower bound of index was done by Choe [1] and Nayatani [9]. Nayatani [10] studied for the index and the nullity of the operator Lg associated to any meromorphic function g on a compact Riemann surface M, how they change under a certain deformation gt of g (t is a positive real number). He considered the derivative ℘ ′ of the Weierstrass ℘-function corresponding to the square lattice Z ⊕ iZ as a meromorphic function g, and computed the index of gt when t is sufficiently small and the nullity of gt for all t. In particular, he showed that there are two values t1, t2(t1 < t2) of t such that the nullity is 4. Furthermore, he investigated the change of index when t becomes large. He showed that the indices of t1g, t2g are 5, and since t2g is the Gauss map of the Costa surface, he could conclude that the index of the Costa surface is 5.