Communications in Analysis and Geometry

Volume 26 (2018)

Number 3

The Gauss–Bonnet formula for harmonic surfaces

Pages: 531 – 570

DOI: http://dx.doi.org/10.4310/CAG.2018.v26.n3.a3

Authors

Peter Connor (Department of Mathematical Sciences, Indiana University, South Bend, In., U.S.A.)

Kevin Li (Department of Computer Science and Mathematical Sciences, Pennsylvania State University Harrisburg, Middletown, Penn., U.S.A.)

Matthias Weber (Department of Mathematical Sciences, Indiana University, South Bend, In., U.S.A.)

Abstract

We consider harmonic immersions in $\mathbb{R}^d$ of compact Riemann surfaces with finitely many punctures where the harmonic coordinate functions are given as real parts of meromorphic functions. We prove that such surfaces have finite total Gauss curvature. The contribution of each end is a multiple of $2 \pi$, determined by the maximal pole order of the meromorphic functions. This generalizes the well known Gackstatter–Jorge–Meeks formula for minimal surfaces. The situation is complicated as the ends with their induced metrics are generally not conformally equivalent to punctured disks, nor do the surfaces generally have limit tangent planes at the ends.

2010 Mathematics Subject Classification

Primary 53C43. Secondary 53C45.

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This work was partially supported by a Simons Foundation grant (no. 246039) to Matthias Weber.

Received 15 March 2014