American Institute of Mathematical Sciences

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  2022, 18: 149-160. doi: 10.3934/jmd.2022007

Hodge and Teichmüller

Jeremy Kahn 1, and  Alex Wright 2,

1. 

Department of Mathematics, Brown University, 151, Thayer Street, Providence, RI 02912, USA
2. 

Department of Mathematics, University of Michigan, 530, Church Street, Ann Arbor, MI 48109, USA

Received  July 16, 2021 Revised  November 22, 2021 Published  April 2022

We consider the derivative $ D\pi $ of the projection $ \pi $ from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form $ \eta $ determines a relative cohomology class $ [\eta]_\Sigma $, which is a tangent vector to the stratum. We give an integral formula for the pairing of $ D\pi([\eta]_\Sigma) $ with a cotangent vector to Teichmüller space (a quadratic differential). We derive from this a comparison between Hodge and Teichmüller norms, which has been used in the work of Arana-Herrera on effective dynamics of mapping class groups, and which may clarify the relationship between dynamical and geometric hyperbolicity results in Teichmüller theory.

Keywords: Hodge norm, Teichmüller metric, strata of Abelian differentials.
Mathematics Subject Classification: Primary: 32G15; Secondary: 30F60, 37D40.
Citation: Jeremy Kahn, Alex Wright. Hodge and Teichmüller. Journal of Modern Dynamics, 2022, 18: 149-160. doi: 10.3934/jmd.2022007
References:
[1]

F. Arana-Herrera, Effective mapping class group dynamics I: Counting lattice points in Teichmüller space, preprint, arXiv: 2010.03123, 2020.

[2]

F. Arana-Herrera, Effective mapping class group dynamics II: Geometric intersection numbers, preprint, arXiv: 2104.01694, 2021.

[3]

F. Arana-Herrera, Effective Mapping Class Group Dynamics III: Counting Filling Closed Curves on Surfaces, preprint, arXiv: 2106.11386, 2021.

[4]

J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J., 161 (2012), 1055-1111.  doi: 10.1215/00127094-1548443.

[5]

M. Bertola, D. Korotkin and C. Norton, Symplectic geometry of the moduli space of projective structures in homological coordinates, Invent. Math., 210 (2017), 759-814.  doi: 10.1007/s00222-017-0739-z.

[6]

K. Burns, H. Masur and A. Wilkinson, The Weil-Petersson geodesic flow is ergodic, Ann. of Math., 175 (2012), 835-908.  doi: 10.4007/annals.2012.175.2.8.

[7]

L. Carleson, P. W. Jones and J.-C. Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. (N.S.), 25 (1994), 1-30.  doi: 10.1007/BF01232933.

[8]

A. Douady and J. Hubbard, On the density of Strebel differentials, Invent. Math., 30 (1975), 175-179.  doi: 10.1007/BF01425507.

[9]

A. Eskin, C. T. McMullen, R. E. Mukamel and A. Wright, Billiards, quadrilaterals, and moduli spaces, J. Amer. Math. Soc., 33 (2020), 1039-1086.  doi: 10.1090/jams/950.

[10]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL$(2, \Bbb R)$ action on moduli space, Publ. Math. Inst. Hautes Études Sci., 127 (2018), 95–324. doi: 10.1007/s10240-018-0099-2.

[11]

A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the SL$(2, \Bbb R)$ action on moduli space, Ann. of Math., 182 (2015), 673-721.  doi: 10.4007/annals.2015.182.2.7.

[12]

A. Eskin, M. Mirzakhani and K. Rafi, Counting closed geodesics in strata, Invent. Math., 215 (2019), 535-607.  doi: 10.1007/s00222-018-0832-y.

[13]

S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, Invent. Math., 205 (2016), 617-670.  doi: 10.1007/s00222-015-0643-3.

[14]

S. Filip, Splitting mixed Hodge structures over affine invariant manifolds, Ann. of Math., 183 (2016), 681-713.  doi: 10.4007/annals.2016.183.2.5.

[15]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math., 155 (2002), 1-103.  doi: 10.2307/3062150.

[16]

G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, J. Mod. Dyn., 8 (2014), 271-436.  doi: 10.3934/jmd.2014.8.271.

[17]

I. Frankel, Meromorphic ${L}^2$ functions on flat surfaces, preprint, arXiv: 2005.13851, 2020.

[18]

É. Goujard, Sous-Variétés Totalement Géodésiques des Espaces de Modules de Riemann, https://www.bourbaki.fr/TEXTES/Exp1178-Goujard.pdf, 2021.

[19]

J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274.  doi: 10.1007/BF02395062.

[20]

M. Kontsevich, Lyapunov exponents and Hodge theory, The Mathematical Beauty of Physics (Saclay, 1996) Adv. Ser. Math. Phys., World Sci. Publ., River Edge, NJ, 24 (1997), 318–332.

[21]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678.  doi: 10.1007/s00222-003-0303-x.

[22]

H. Masur, The Teichmüller flow is Hamiltonian, Proc. Amer. Math. Soc., 123 (1995), 3739-3747.  doi: 10.2307/2161902.

[23]

C. T. McMullen, Navigating moduli space with complex twists, J. Eur. Math. Soc. (JEMS), 15 (2013), 1223-1243.  doi: 10.4171/JEMS/390.

[24]

C. T. McMullen, R. E. Mukamel and A. Wright, Cubic curves and totally geodesic subvarieties of moduli space, Ann. of Math., 185 (2017), 957-990.  doi: 10.4007/annals.2017.185.3.6.

[25]

M. Mirzakhani and A. Wright, Full-rank affine invariant submanifolds, Duke Math. J., 167 (2018), 1-40.  doi: 10.1215/00127094-2017-0036.

[26]

K. Rafi, Hyperbolicity in Teichmüller space, Geom. Topol., 18 (2014), 3025-3053.  doi: 10.2140/gt.2014.18.3025.

[27]

S. A. Wolpert, Schiffer variations and Abelian differentials, Adv. Math., 333 (2018), 497-522.  doi: 10.1016/j.aim.2018.05.019.

[28]

A. Wright, Totally geodesic submanifolds of Teichmüller space, J. Differential Geom., 115 (2020), 565-575.  doi: 10.4310/jdg/1594260019.

show all references

References:
[1]

F. Arana-Herrera, Effective mapping class group dynamics I: Counting lattice points in Teichmüller space, preprint, arXiv: 2010.03123, 2020.

[2]

F. Arana-Herrera, Effective mapping class group dynamics II: Geometric intersection numbers, preprint, arXiv: 2104.01694, 2021.

[3]

F. Arana-Herrera, Effective Mapping Class Group Dynamics III: Counting Filling Closed Curves on Surfaces, preprint, arXiv: 2106.11386, 2021.

[4]

J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J., 161 (2012), 1055-1111.  doi: 10.1215/00127094-1548443.

[5]

M. Bertola, D. Korotkin and C. Norton, Symplectic geometry of the moduli space of projective structures in homological coordinates, Invent. Math., 210 (2017), 759-814.  doi: 10.1007/s00222-017-0739-z.

[6]

K. Burns, H. Masur and A. Wilkinson, The Weil-Petersson geodesic flow is ergodic, Ann. of Math., 175 (2012), 835-908.  doi: 10.4007/annals.2012.175.2.8.

[7]

L. Carleson, P. W. Jones and J.-C. Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. (N.S.), 25 (1994), 1-30.  doi: 10.1007/BF01232933.

[8]

A. Douady and J. Hubbard, On the density of Strebel differentials, Invent. Math., 30 (1975), 175-179.  doi: 10.1007/BF01425507.

[9]

A. Eskin, C. T. McMullen, R. E. Mukamel and A. Wright, Billiards, quadrilaterals, and moduli spaces, J. Amer. Math. Soc., 33 (2020), 1039-1086.  doi: 10.1090/jams/950.

[10]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL$(2, \Bbb R)$ action on moduli space, Publ. Math. Inst. Hautes Études Sci., 127 (2018), 95–324. doi: 10.1007/s10240-018-0099-2.

[11]

A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the SL$(2, \Bbb R)$ action on moduli space, Ann. of Math., 182 (2015), 673-721.  doi: 10.4007/annals.2015.182.2.7.

[12]

A. Eskin, M. Mirzakhani and K. Rafi, Counting closed geodesics in strata, Invent. Math., 215 (2019), 535-607.  doi: 10.1007/s00222-018-0832-y.

[13]

S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, Invent. Math., 205 (2016), 617-670.  doi: 10.1007/s00222-015-0643-3.

[14]

S. Filip, Splitting mixed Hodge structures over affine invariant manifolds, Ann. of Math., 183 (2016), 681-713.  doi: 10.4007/annals.2016.183.2.5.

[15]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math., 155 (2002), 1-103.  doi: 10.2307/3062150.

[16]

G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, J. Mod. Dyn., 8 (2014), 271-436.  doi: 10.3934/jmd.2014.8.271.

[17]

I. Frankel, Meromorphic ${L}^2$ functions on flat surfaces, preprint, arXiv: 2005.13851, 2020.

[18]

É. Goujard, Sous-Variétés Totalement Géodésiques des Espaces de Modules de Riemann, https://www.bourbaki.fr/TEXTES/Exp1178-Goujard.pdf, 2021.

[19]

J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274.  doi: 10.1007/BF02395062.

[20]

M. Kontsevich, Lyapunov exponents and Hodge theory, The Mathematical Beauty of Physics (Saclay, 1996) Adv. Ser. Math. Phys., World Sci. Publ., River Edge, NJ, 24 (1997), 318–332.

[21]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678.  doi: 10.1007/s00222-003-0303-x.

[22]

H. Masur, The Teichmüller flow is Hamiltonian, Proc. Amer. Math. Soc., 123 (1995), 3739-3747.  doi: 10.2307/2161902.

[23]

C. T. McMullen, Navigating moduli space with complex twists, J. Eur. Math. Soc. (JEMS), 15 (2013), 1223-1243.  doi: 10.4171/JEMS/390.

[24]

C. T. McMullen, R. E. Mukamel and A. Wright, Cubic curves and totally geodesic subvarieties of moduli space, Ann. of Math., 185 (2017), 957-990.  doi: 10.4007/annals.2017.185.3.6.

[25]

M. Mirzakhani and A. Wright, Full-rank affine invariant submanifolds, Duke Math. J., 167 (2018), 1-40.  doi: 10.1215/00127094-2017-0036.

[26]

K. Rafi, Hyperbolicity in Teichmüller space, Geom. Topol., 18 (2014), 3025-3053.  doi: 10.2140/gt.2014.18.3025.

[27]

S. A. Wolpert, Schiffer variations and Abelian differentials, Adv. Math., 333 (2018), 497-522.  doi: 10.1016/j.aim.2018.05.019.

[28]

A. Wright, Totally geodesic submanifolds of Teichmüller space, J. Differential Geom., 115 (2020), 565-575.  doi: 10.4310/jdg/1594260019.

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