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Bolza Surface

The Bolza surface is the closed hyperbolic Riemann surface of genus 2 obtained by identifying opposite sides of a regular octagon in the hyperbolic plane with interior angles pi/4 (Katz and Sabourau 2006). Its hyperbolic metric has constant Gaussian curvature -1. The regular octagon is hyperbolic, rather than a Euclidean regular octagon with its flat metric.

Among closed hyperbolic surfaces of genus 2, the Bolza surface maximizes the systole, the length of a shortest noncontractible closed geodesic. Matsuda and Yamazaki (2026) proved that it does not locally minimize the intrinsic generalized diameter on Teichmüller space. Thus arbitrarily small deformations can decrease its generalized diameter, despite its extremal systole.

See also

Hyperbolic Plane, Riemann Surface, Systole, Teichmüller Space

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References

Katz, M. G. and Sabourau, S. "An Optimal Systolic Inequality for CAT(0) Metrics in Genus Two." Pacific J. Math. 227, 95-107, 2006. https://doi.org/10.2140/pjm.2006.227.95.Matsuda, R. and Yamazaki, K. "The Bolza Surface Does Not Locally Minimize the Diameter Function." 12 Nov 2026. https://arxiv.org/abs/2611.16340v1.

Cite this as:

Weisstein, Eric W. "Bolza Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BolzaSurface.html

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