## Duke Mathematical Journal

### Schwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volume

#### Abstract

To a complex projective structure $\Sigma$ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms $\|\phi _{\Sigma }\|_{\infty }$ and $\|\phi _{\Sigma }\|_{2}$ of the quadratic differential $\phi _{\Sigma }$ of $\Sigma$ given by the Schwarzian derivative of the associated locally univalent map. We show that these give a unifying approach that generalizes a number of important, well-known results for convex cocompact hyperbolic structures on $3$-manifolds, including bounds on the Lipschitz constant for the nearest-point retraction and the length of the bending lamination. We then use these bounds to begin a study of the Weil–Petersson gradient flow of renormalized volume on the space $\operatorname{CC}(N)$ of convex cocompact hyperbolic structures on a compact manifold $N$ with incompressible boundary, leading to a proof of the conjecture that the renormalized volume has infimum given by one half the simplicial volume of $\mathit{DN}$, the double of $N$.

#### Article information

Source
Duke Math. J., Volume 168, Number 5 (2019), 867-896.

Dates
Revised: 15 November 2018
First available in Project Euclid: 2 March 2019

https://projecteuclid.org/euclid.dmj/1551495708

Digital Object Identifier
doi:10.1215/00127094-2018-0061

Mathematical Reviews number (MathSciNet)
MR3934591

Zentralblatt MATH identifier
07055195

#### Citation

Bridgeman, Martin; Brock, Jeffrey; Bromberg, Kenneth. Schwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volume. Duke Math. J. 168 (2019), no. 5, 867--896. doi:10.1215/00127094-2018-0061. https://projecteuclid.org/euclid.dmj/1551495708

#### References

• [1] C. G. Anderson, Projective structures on Riemann surfaces and developing maps to $\mathbb{H}^{3}$ and $\mathbb{CP}^{n}$, Ph.D. thesis, University of California, Berkeley, 1998.
• [2] C. J. Bishop, Quasiconformal Lipschitz maps, Sullivan’s convex hull theorem and Brennan’s conjecture, Ark. Mat. 40 (2002), no. 1, 1–26.
• [3] M. Bridgeman, J. F. Brock, and K. W. Bromberg, Schwarzian derivatives and renormalized volume II: the Weil–Petersson gradiant flow and volumes of hyperbolic convex cores, preprint, 2017.
• [4] M. Bridgeman and R. D. Canary, Bounding the bending of a hyperbolic $3$-manifold, Pacific J. Math. 218 (2005), no. 2, 299–314.
• [5] M. Bridgeman and R. D. Canary, The Thurston metric on hyperbolic domains and boundaries of convex hulls, Geom. Funct. Anal. 20 (2010), no. 6, 1317–1353.
• [6] M. Bridgeman and R. D. Canary, Renormalized volume and volume of the convex core, Ann. Inst. Fourier 67 (2017), no. 5, 2083–2098.
• [7] J. Brock, The Weil–Petersson metric and volumes of $3$-dimensional hyperbolic convex cores, J. Amer. Math. Soc. 16 (2003), no. 3, 495–535.
• [8] J. Brock, Weil–Petersson translation distance and volumes of mapping tori, Comm. Anal. Geom. 11 (2003), no. 5, 987–999.
• [9] K. Bromberg, Hyperbolic cone-manifolds, short geodesics, and Schwarzian derivatives, J. Amer. Math. Soc. 17 (2004), no. 4, 783–826.
• [10] R. D. Canary, The Poincaré metric and a conformal version of a theorem of Thurston, Duke Math. J. 64 (1991), no. 2, 349–359.
• [11] C. Ciobotaru and S. Moroianu, Positivity of the renormalized volume of almost-Fuchsian hyperbolic $3$-manifolds, Proc. Amer. Math. Soc. 144 (2016), no. 1, 151–159.
• [12] C. Epstein, Envelopes of horospheres and Weingarten surfaces in hyperbolic $3$-spaces, preprint, 1984, https://www.math.upenn.edu/~cle/papers/WeingartenSurfaces.pdf.
• [13] D. B. A. Epstein, A. Marden, and V. Markovic, Quasiconformal homeomorphisms and the convex hull boundary, Ann. of Math. (2) 159 (2004), no. 1, 305–336.
• [14] C. R. Graham and E. Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nuclear Phys. B 546 (1999), no. 1–2, 52–64.
• [15] 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.
• [16] I. Kra and B. Maskit, “Remarks on projective structures” in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud. 97, Princeton Univ. Press, Princeton, 1981, 343–359.
• [17] K. Krasnov and J.-M. Schlenker, On the renomalized volume of hyperbolic $3$-manifolds, Comm. Math. Phys. 279 (2008), no. 3, 637–668.
• [18] H. A. Masur and Y. N. Minsky, Geometry of the complex of curves, I: Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149.
• [19] H. A. Masur and Y. N. Minsky, Geometry of the complex of curves, II: Hierarchical structure, Geom. Funct. Anal. 10 (2000), no. 4, 902–974.
• [20] S. Moroianu, Convexity of the renormalized volume of hyperbolic $3$-manifolds, Amer. J. Math. 139 (2017), no. 5, 1379–1394.
• [21] Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551.
• [22] F. V. Pallete, Local convexity for renormalized volume for rank-$1$ cusped manifolds, to appear in Math. Res. Lett.
• [23] F. V. Pallete, Continuity of renormalized volume under geometric limits, preprint, arXiv:1605.07986 [math.DG].
• [24] J.-M. Schlenker, The renormalized volume and the volume of the convex core of quasifuchsian manifolds, Math. Res. Lett. 20 (2013), no. 4, 773–786.
• [25] P. A. Storm, Minimal volume Alexandrov spaces, J. Differential Geom. 61 (2002), no. 2, 195–225.
• [26] L. A. Takhtajan and L.-P. Teo, Liouville action and Weil–Petersson metric on deformation spaces, global Kleinian reciprocity and holography, Comm. Math. Phys. 239 (2003), no. 1–2, 183–240.
• [27] P. G. Zograf and L. A. Takhtadzhyan, On the uniformization of Riemann surfaces and on the Weil–Petersson metric on the Teichmüller and Schottky spaces (in Russian), Mat. Sb. (N.S.) 132(174), no. 3 (1987), 304–321, 444; English translation in Math. USSR-Sb 60 (1988), no 2, 297–313.