## Duke Mathematical Journal

### The Schwarzian derivative and measured laminations on Riemann surfaces

David Dumas

#### Abstract

A holomorphic quadratic differential on a hyperbolic Riemann surface has an associated measured foliation that can be straightened to yield a measured geodesic lamination. On the other hand, a quadratic differential can be regarded as the Schwarzian derivative of a $\mathbb{CP}^1$-structure, to which one can naturally associate another measured geodesic lamination using grafting.

We compare these two relationships between quadratic differentials and measured geodesic laminations, each of which yields a homeomorphism $\mathscr{M\!L}(S) \to Q(X)$ for each conformal structure $X$ on a compact surface $S$. We show that these maps are nearly identical, differing by a multiplicative factor of $-2$ and an error term of lower order than the maps themselves (which we bound explicitly).

As an application, we show that the Schwarzian derivative of a $\mathbb{CP}^1$-structure with Fuchsian holonomy is close to a $2\pi$-integral Jenkins-Strebel differential. We also study two compactifications of the space of $\mathbb{CP}^1$-structures, one of which uses the Schwarzian derivative and another of which uses grafting coordinates. The natural map between these two compactifications is shown to extend to the boundary of each fiber over Teichmüller space, and we describe that extension

#### Article information

Source
Duke Math. J., Volume 140, Number 2 (2007), 203-243.

Dates
First available in Project Euclid: 18 October 2007

https://projecteuclid.org/euclid.dmj/1192715419

Digital Object Identifier
doi:10.1215/S0012-7094-07-14021-3

Mathematical Reviews number (MathSciNet)
MR2359819

Zentralblatt MATH identifier
1134.30035

#### Citation

Dumas, David. The Schwarzian derivative and measured laminations on Riemann surfaces. Duke Math. J. 140 (2007), no. 2, 203--243. doi:10.1215/S0012-7094-07-14021-3. https://projecteuclid.org/euclid.dmj/1192715419

#### References

• L. V. Ahlfors, Cross-ratios and Schwarzian derivatives in ${\bf R}\sp n$'' in Complex Analysis, Birkhäuser, Basel, 1988, 1--15.
• A. D. Aleksandrov, V. N. Berestovskiĭ, and I. G. Nikolaev, Generalized Riemannian spaces (in Russian), Uspekhi Mat. Nauk 41, no. 3 (1986), 3--44.; English translation in Russian Math. Surveys 41 (1986), 1--54.
• C. G. Anderson, Projective structures on Riemann surfaces and developing maps to $\mathbb{H}^3$ and $\mathbb{CP}^n$, Ph.D. dissertation, University of California, Berkeley, Berkeley, Calif., 1998.
• S. Bouarroudj and V. Yu. Ovsienko, Three cocycles on Diff$(S^1)$ generalizing the Schwarzian derivative, Internat. Math. Res. Notices 1998, no. 1, 25--39.
• A. Brudnyi, On a BMO-property for subharmonic functions, J. Fourier Anal. Appl. 8 (2002), 603--612.
• M. Chuaqui, P. Duren, and B. Osgood, The Schwarzian derivative for harmonic mappings, J. Anal. Math. 91 (2003), 329--351.
• —, Curvature properties of planar harmonic mappings, Comput. Methods Funct. Theory 4 (2004), 127--142.
• M. Culler and P. B. Shalen, Varieties of group representations and splittings of $3$-manifolds, Ann. of Math. (2) 117 (1983), 109--146.
• D. Dumas, Grafting, pruning, and the antipodal map on measured laminations, J. Differential Geom. 74 (2006), 93--118.; Erratum, J. Differential Geom. 77 (2007), 175--176.
• —, \removethiszBear: A tool for studying Bers slices of punctured tori, http://bear.sourceforge.net/
• K. D'Yuval' [C. Duval] and V. Yu. Ovsienko, Lorentz world lines and the Schwarzian derivative (in Russian), Funktsional. Anal. i Prilozhen. 34, no. 2 (2000), 69--72.; English translation in Funct. Anal. Appl. 34 (2000), 135--137.
• C. L. Epstein, The hyperbolic Gauss map and quasiconformal reflections, J. Reine Angew. Math. 372 (1986), 96--135.
• H. Flanders, The Schwarzian as a curvature, J. Differential Geometry 4 (1970), 515--519.
• D. Gallo, M. Kapovich, and A. Marden, The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math. (2) 151 (2000), 625--704.
• F. P. Gardiner, The existence of Jenkins-Strebel differentials from Teichmüller theory, Amer. J. Math. 99 (1977), 1097--1104.
• —, Measured foliations and the minimal norm property for quadratic differentials, Acta Math. 152 (1984), 57--76.
• D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss. 224, Springer, Berlin, 1983.
• W. M. Goldman, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), 297--326.
• S. Gong and C. H. Fitzgerald, The Schwarzian derivative in several complex variables, Sci. China Ser. A 36 (1993), 513--523.
• R. C. Gunning, Special coordinate coverings of Riemann surfaces, Math. Ann. 170 (1967), 67--86.
• —, Affine and projective structures on Riemann surfaces'' in Riemann Surfaces and Related Topics (Stony Brook, N.Y., 1978), Ann. of Math. Stud. 97, Princeton Univ. Press, Princeton, 1981, 225--244.
• D. A. Hejhal, Monodromy groups and linearly polymorphic functions, Acta Math. 135 (1975), 1--55.
• J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), 221--274.
• A. Huber, Zum potentialtheoretischen Aspekt der Alexandrowschen Flächentheorie, Comment. Math. Helv. 34 (1960), 99--126.
• J. A. Jenkins, On the existence of certain general extremal metrics, Ann. of Math. (2) 66 (1957), 440--453.
• Y. Kamishima and S. P. Tan, Deformation spaces on geometric structures'' in Aspects of Low-Dimensional Manifolds, Adv. Stud. Pure Math. 20, Kinokuniya, Tokyo, 1992, 263--299.
• M. Kapovich, Hyperbolic Manifolds and Discrete Groups, Progr. Math. 183, Birkhäuser, Boston, 2001.
• L. Keen and C. Series, Pleating invariants for punctured torus groups, Topology 43 (2004), 447--491.
• S. P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980), 23--41.
• S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, 2nd ed., World Sci., Hackensack, N.J., 2005.
• Y. Komori and T. Sugawa, Bers embedding of the Teichmüller space of a once-punctured torus, Conform. Geom. Dyn. 8 (2004), 115--142.
• Y. Komori, T. Sugawa, M. Wada, and Y. Yamashita, Drawing Bers embeddings of the Teichmüller space of once-punctured tori, Experiment. Math. 15 (2006), 51--60.
• N. J. Korevaar and R. M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), 561--659.
• I. Kra, A generalization of a theorem of Poincaré, Proc. Amer. Math. Soc. 27 (1971), 299--302.
• R. S. Kulkarni and U. Pinkall, A canonical metric for Möbius structures and its applications, Math. Z. 216 (1994), 89--129.
• G. Levitt, Foliations and laminations on hyperbolic surfaces, Topology 22 (1983), 119--135.
• A. Marden and K. Strebel, The heights theorem for quadratic differentials on Riemann surfaces, Acta Math. 153 (1984), 153--211.
• B. Maskit, On a class of Kleinian groups, Ann. Acad. Sci. Fenn. Ser. A I 442, (1969).
• C. Mese, The structure of singular spaces of dimension $2$, Manuscripta Math. 100 (1999), 375--389.
• —, The curvature of minimal surfaces in singular spaces, Comm. Anal. Geom. 9 (2001), 3--34.
• —, Harmonic maps between surfaces and Teichmüller spaces, Amer. J. Math. 124 (2002), 451--481.
• Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545--551.
• B. Osgood and D. Stowe, The Schwarzian derivative and conformal mapping of Riemannian manifolds, Duke Math. J. 67 (1992), 57--99.
• Ju. G. Rešetnjak [yu. g. reshetnyak], Isothermal coordinates on manifolds of bounded curvature, I, II (in Russian), Sibirsk. Mat. Ž. 1 (1960), 88--116.; 248--276.
• —, On the conformal representation of Alexandrov surfaces'' in Papers on Analysis, Rep. Univ. Jyväskylä Dep. Math. Stat. 83, Univ. Jyväskylä, Jyväskylä, Finland, 2001, 287--304.
• K. P. Scannell and M. Wolf, The grafting map of Teichmüller space, J. Amer. Math. Soc. 15 (2002), 893--927.
• H. Shiga, Projective structures on Riemann surfaces and Kleinian groups, J. Math. Kyoto Univ. 27 (1987), 433--438.
• H. Shiga and H. Tanigawa, Projective structures with discrete holonomy representations, Trans. Amer. Math. Soc. 351 (1999), no. 2, 813--823.
• K. Strebel, Über quadratische Differentiate mit geschlossenen Trajektorien und extremale quasikonforme Abbildungen'' in Festband zum 70 (Zurich, 1965), Springer, Berlin, 1966, 105--127.
• —, Quadratic Differentials, Ergeb. Math. Grenzgeb. (3) 5, Springer, Berlin, 1984.
• H. Tanigawa, Grafting, harmonic maps and projective structures on surfaces, J. Differential Geom. 47 (1997), 399--419.
• —, Divergence of projective structures and lengths of measured laminations, Duke Math. J. 98 (1999), 209--215.
• W. P. Thurston, Zippers and univalent functions'' in The Bieberbach Conjecture (West Lafayette, Ind., 1985), Math. Surveys Monogr. 21, Amer. Math. Soc., Providence, 1986, 185--197.
• M. Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), 449--479.
• —, High energy degeneration of harmonic maps between surfaces and rays in Teichmüller space, Topology 30 (1991), 517--540.
• —, Harmonic maps from surfaces to $\bold R$-trees, Math. Z. 218 (1995), 577--593.
• —, On the existence of Jenkins-Strebel differentials using harmonic maps from surfaces to graphs, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), 269--278.
• —, On realizing measured foliations via quadratic differentials of harmonic maps to $\bold R$-trees, J. Anal. Math. 68 (1996), 107--120.