TY - JOUR

T1 - The asymptotic behavior of the monodromy representations of the associated families of compact CMC surfaces

AU - Heller, Sebastian

N1 - Funding information: S. Heller was supported by DFG HE 6829/1-2.

PY - 2015/5/4

Y1 - 2015/5/4

N2 - Constant mean curvature (CMC) surfaces in space forms can be described by their associated \(\mathbb C^*\)-family of flat \(SL(2,\mathbb C)\)-connections \(\nabla^\lambda\). In this paper we consider the asymptotic behavior (for \(\lambda\to0\)) of the gauge equivalence classes of \(\nabla^\lambda\) for compact CMC surfaces of genus \(g\geq2.\) We prove (under the assumption of simple umbilics) that the asymptotic behavior of the traces of the monodromy representation of \(\nabla^{\lambda}\) determines the conformal type as well as the Hopf differential locally in the Teichm\"uller space.

AB - Constant mean curvature (CMC) surfaces in space forms can be described by their associated \(\mathbb C^*\)-family of flat \(SL(2,\mathbb C)\)-connections \(\nabla^\lambda\). In this paper we consider the asymptotic behavior (for \(\lambda\to0\)) of the gauge equivalence classes of \(\nabla^\lambda\) for compact CMC surfaces of genus \(g\geq2.\) We prove (under the assumption of simple umbilics) that the asymptotic behavior of the traces of the monodromy representation of \(\nabla^{\lambda}\) determines the conformal type as well as the Hopf differential locally in the Teichm\"uller space.

KW - math.DG

KW - 53A10, 53C42, 53C43

U2 - 10.1112/blms/bdw036

DO - 10.1112/blms/bdw036

M3 - Article

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

SN - 0024-6093

ER -