Uniformization of branched surfaces and Higgs bundles. (English) Zbl 1483.30078
Summary: Given a compact connected Riemann surface \(\Sigma\) of genus \(g_{\Sigma}\geq 2\), and an effective divisor \(D=\sum_i n_i x_i\) on \(\Sigma\) with \(\mathrm{degree}(D)<2( g_{\Sigma}-1)\), there is a unique cone metric on \(\Sigma\) of constant negative curvature \(-4\) such that the cone angle at each point \(x_i\) is \(2\pi n_i\)
[R. C. McOwen, Proc. Am. Math. Soc. 103, No. 1, 222–224 (1988; Zbl 0657.30033); M. Troyanov, Trans. Am. Math. Soc. 324, No. 2, 793–821 (1991; Zbl 0724.53023)].
We describe the Higgs bundle on \(\Sigma\) corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on \(\Sigma\) parametrized by a nonempty open subset of \(H^0(\Sigma, K_{\Sigma}^{\otimes 2}\otimes\mathcal{O}_{\Sigma}(-2D))\) that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin’s results in
[N. J. Hitchin, Proc. Lond. Math. Soc. (3) 55, 59–126 (1987; Zbl 0634.53045)]
for the case \(D=0\).
MSC:
| 30F10 | Compact Riemann surfaces and uniformization |
| 14H60 | Vector bundles on curves and their moduli |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 53C43 | Differential geometric aspects of harmonic maps |
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