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Let $X$ be a Riemann surface with analytic boundary. Assume that $X$ has negative Euler characteristic. Then there exists a conformal hyperbolic metric $X$ such that $\partial X$ consists of geodesics (conformal in the sense of being compatible with the given complex structure on $X$).

Does anyone has a reference (or even better, a quick proof) of this result?

Edit: Let me state a stronger version. Suppose $X$ is embedded in an open Riemann surface $Y$. Then there exists a conformal hyperbolic metric on a neighborhood of $X$ in $Y$ such that $\partial X$ consists of geodesics. Is this true?

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  • $\begingroup$ The 1992 book: P. Buser, Geometry and Spectra of Compact Riemann Surfaces? $\endgroup$
    – YCor
    Commented May 10, 2023 at 15:07
  • $\begingroup$ The stronger version also holds: actually a neighborhood N of X (in the doubling of X) also embeds into Y. Indeed by the Schwarz reflection principle the embedding of X into Y extends to a holomorphic map from a neighborhood N into Y. This extension is an embedding, if N is sufficiently small. $\endgroup$
    – user_1789
    Commented May 10, 2023 at 18:45

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A good reference is

W. Abikoff, The real analytic theory of Teichmuller space, Springer, 1980. (Chap. II section 1).

The idea is that you construct the double: it is the result of gluing of your surface with its mirror image. This is a compact surface, it has a hyperbolic metric, and the restriction of this metric on the original surface is the hyperbolic metric with geodesic boundary.

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See Theorem 1(b) in the following paper; it is enough that the boundary is smooth.

Osgood, B.; Phillips, R.; Sarnak, P. Extremals of determinants of Laplacians. J. Funct. Anal. 80 (1988), no. 1, 148--211.

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  • $\begingroup$ But analyticity of the boundary is necessary. Consider, e.g., a pair of pants. If every pair of pants admits such a metric, then they would be isomorphic to a standard hyperbolic pair of pants, which already has analytic boundary. I guess a more precise way of stating what I'm looking for is that the hyperbolic metric should extend to a hyperbolic metric in a neighborhood of $X$ (after embedding $X$ in an open Riemann surface). $\endgroup$
    – Yuxiao Xie
    Commented May 10, 2023 at 7:23
  • $\begingroup$ @YuxiaoXie: You are wrong (confusing different categories: Your question was in differential geometry category and now you are switching to a related, but different category of bordered Riemann surfaces). $\endgroup$
    – Moishe Kohan
    Commented May 10, 2023 at 15:23
  • $\begingroup$ @MoisheKohan I don't understand why my question was in the differential category. I work on a given Riemann surface and I only consider conformal metrics on it. $\endgroup$
    – Yuxiao Xie
    Commented May 10, 2023 at 16:30

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