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

2021/04/19 by Sarenhu
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.2104.09013

arxiv created 2021/04/19 · arxiv updated 2021/04/20

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, G \colon M → S² is the holomorphic map corresponding to g. We consider the meromorphic function w on the Riemann surface M_a= \(z,w) ∈\widehatℂ² | w²=z(z-a)(z+(1)/(a))\(a \geqslant 1 ) homeomorphic to a torus, and we determine the index of tw for all a in the range 1 \leqslant a \leqslant a₀ (where a₀ can be numerically evaluated) and all t>0.

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