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

@article{Bridgeman2017SchwarzianDP,
title={Schwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volume},
author={Martin J. Bridgeman and Jeffrey Brock and Kenneth W. Bromberg},
journal={Duke Mathematical Journal},
year={2017},
volume={168},
pages={867-896}
}
• Published 2017
• Mathematics
• Duke Mathematical Journal
• 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… CONTINUE READING
3

#### Citations

##### Publications citing this paper.
SHOWING 1-10 OF 10 CITATIONS

## The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores

• Mathematics
• 2020

## Notes on the Schwarzian tensor and measured foliations at infinity of quasifuchsian manifolds.

VIEW 3 EXCERPTS
CITES BACKGROUND

## Volumes of quasifuchsian manifolds

VIEW 2 EXCERPTS
CITES BACKGROUND

## Comparing hyperbolic and extremal lengths for shortest curves

• Mathematics
• 2019

## CMC-1 surfaces via osculating M\"{o}bius transformations between circle patterns

#### References

##### Publications referenced by this paper.
SHOWING 1-10 OF 30 REFERENCES

## ON UNIFORMIZATION OF RIEMANN SURFACES AND THE WEIL-PETERSSON METRIC ON TEICHMÜLLER AND SCHOTTKY SPACES

• Mathematics
• 1988

## Convexity of the Renormalized volume of hyperbolic 3-manifolds

VIEW 1 EXCERPT

## The Thurston Metric on Hyperbolic Domains and Boundaries of Convex Hulls

• Mathematics
• 2010

## Quasiconformal homeomorphisms and the convex hull boundary

• Mathematics
• 2004

## Continuity of the renormalized volume under geometric limits

VIEW 6 EXCERPTS
HIGHLY INFLUENTIAL

## Liouville Action and Weil-Petersson Metric on Deformation Spaces, Global Kleinian Reciprocity and Holography

• Mathematics, Physics
• 2002

## Geometry of the complex of curves I: Hyperbolicity

• Mathematics
• 1999

## Quasiconformal Lipschitz maps, Sullivan's convex hull theorem and Brennan's conjecture

VIEW 1 EXCERPT