Index and Nullity of the Gauss Map of the Costa-hoffman-meeks Surfaces

The aim of this work is to extend the results of S. Nayatani about the index and the nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces for values of the genus bigger than 37. That allows us to state that these minimal surfaces are non degenerate for all the values of the genus in the sense of the definition of J. Pérez and A. Ros. Introduction In the years 80’s and 90’s the study of the index of minimal surfaces in Euclidean space has been quite active. D. Fischer-Colbrie in [4], R. Gulliver and H. B. Lawson in [5] proved independently that a complete minimal surface M in R with Gauss map G has finite index if and only if it has finite total curvature. D. Fischer-Colbrie also observed that if M has finite total curvature its index coincides with the index of an operator LḠ (that is the number of its negative eigenvalues) associated to the extended Gauss map Ḡ of M̄, the compactification of M. Moreover N(Ḡ), the null space of LḠ, if restricted to M consists of the bounded solutions of the Jacobi equation. The nullity, Nul(Ḡ), that is the dimension of N(Ḡ), and the index are invariants of Ḡ because they are independent of the choice of the conformal metric on M̄. The computation of the index and of the nullity of the Gauss map of the Costa surface and of the Costa-Hoffman-Meeks surface of genus g = 2, . . . , 37 appeared respectively in the works [10] and [9] of S. Nayatani. The aim of this work is to extend his results to the case where g > 38. In [10] he studied the index and the nullity of the operator LG associated to an arbitrary holomorphic map G : Σ → S, where Σ is a compact Riemann surface. He considered a deformation Gt : Σ → S, t ∈ (0,+∞), with G1 = G (see equation (2)) and gave lower and upper bounds for the index of Gt, Ind(Gt), and its nullity, Nul(Gt), for t near to 0 and +∞ and t = 1. The computation of the index and the nullity in the case of the Costa surface is based on the fact that the Gauss map of this surface is a deformation for a particular value of t of the map G defined by π ◦G = 1/℘′, that is its stereographic projection is equal to the inverse of the derivative of the Weierstrass ℘-function for a unit 2000 Mathematics Subject Classification. 58E12, 49Q05, 53A10.