×

Plurisubharmonicity and geodesic convexity of energy function on Teichmüller space. (English) Zbl 1502.30132

The classical Teichmüller space carries different metrics and energy functions defined by them. In this paper the authors study the convexity of a particular function defined via the Weil-Petersson metric.
For a Riemannian manifold \(M^n\) and a fixed Riemannian surface \(X(z)\) with a hyperbolic metric \(\Phi(z)\) with \(z\) a point in the Teichmüller space of Riemannian surfaces of genus \(g\), it is known that any continuous map \(u_0 : M\rightarrow X(z)\) is homotopic to a unique harmonic map \(u\) unless the image is a point or a closed geodesic. The energy \(E(u)\) of \(u\) is a function of \(z\), so defines a function \(E(z)\) on the Teichmüller space. The main result of the paper is that \(\log E(z)\) is plurisubharmonic. Previously similar property has been shown for the function defining the energy of the harmonic map in a homotopy class of maps \(u_0: X(z)\rightarrow M\) in [D. Toledo, Geom. Funct. Anal. 22, No. 4, 1015–1032 (2012; Zbl 1254.32020)].
The main result is proven by careful study of the first and second variation of the energy function. As a corollaries the authors obtain another proofs of some results about the length of geodesics function on Teichmüller space.

MSC:

30F60 Teichmüller theory for Riemann surfaces
58E20 Harmonic maps, etc.

Citations:

Zbl 1254.32020
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] L. V. AHLFORS, Some remarks on Teichmüller’s space of Riemann surfaces, Ann. of Math. (2) 74 (1961), 171-191. http://dx.doi.org/10.2307/1970309. MR204641 · Zbl 0146.30602 · doi:10.2307/1970309.MR204641
[2] R. AXELSSON and G. SCHUMACHER, Geometric approach to the Weil-Petersson symplectic form, Comment. Math. Helv. 85 (2010), no. 2, 243-257. http://dx.doi.org/10.4171/CMH/ 194. MR2595178 · Zbl 1193.30065 · doi:10.4171/CMH/194.MR2595178
[3] , Variation of geodesic length functions in families of Kähler-Einstein manifolds and applica-tions to Teichmüller space, Ann. Acad. Sci. Fenn. Math. 37 (2012), no. 1, 91-106. http://dx. doi.org/10.5186/aasfm.2012.3703. MR2920426 · Zbl 0146.30602 · doi:10.2307/1970309
[4] S. I. ALBER, Spaces of mappings into a manifold of negative curvature, Dokl. Akad. Nauk SSSR 178 (1968), 13-16 (Russian). MR0230254 · Zbl 0165.55704
[5] L. BERS and L. EHRENPREIS, Holomorphic convexity of Teichmüller spaces, Bull. Amer. Math. Soc. 70 (1964), 761-764. · Zbl 0136.07004
[6] J. P. DEMAILLY, Complex Analytic And Differential Geometry, 2012. https://www-fourier.ujf-grenoble.fr/demailly/manuscripts/agbook.pdf. · Zbl 1267.53075 · doi:10.5186/aasfm.2012.3703
[7] J. EELLS Jr. and L. LEMAIRE, Deformations of metrics and associated harmonic maps, Geometry and Analysis, Indian Acad. Sci., Bangalore, 1980, pp. 33-45. MR592252 · Zbl 0509.58017
[8] J. EELLS Jr. and J. H. SAMPSON, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. http://dx.doi.org/10.2307/2373037. MR164306 · Zbl 0136.07004 · doi:10.1090/S0002-9904-1964-11230-1
[9] H. FENG, K. LIU, and X. WAN, Geodesic-Einstein metrics and nonlinear stabilities, Trans. Amer. Math. Soc. 371 (2019), no. 11, 8029-8049. http://dx.doi.org/10.1090/tran/ 7658. MR3955542 · Zbl 1415.53058 · doi:10.1090/tran/7658.MR3955542
[10] A. E. FISCHER and J. E. MARSDEN, Deformations of the scalar curvature, Duke Math. J. 42 (1975), no. 3, 519-547. · Zbl 0336.53032
[11] http://dx.doi.org/10.1215/S0012-7094-75-04249-0. MR380907 · Zbl 0122.40102 · doi:10.2307/2373037
[12] P. HARTMAN, On homotopic harmonic maps, Canadian J. Math. 19 (1967), 673-687. http:// dx.doi.org/10.4153/CJM-1967-062-6. MR214004 · Zbl 0148.42404 · doi:10.4153/CJM-1967-062-6.MR214004
[13] J. JOST, Two-dimensional Geometric Variational Problems, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1991. A Wiley-Interscience Publica-tion. MR1100926 · Zbl 1415.53058 · doi:10.1090/tran/7658
[14] N. KOISO, Variation of harmonic mapping caused by a deformation of Riemannian met-ric, Hokkaido Math. J. 8 (1979), no. 2, 199-213. http://dx.doi.org/10.14492/hokmj/ 1381758271. MR551551 · Zbl 0148.42404 · doi:10.4153/CJM-1967-062-6
[15] S. P. KERCKHOFF, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235-265. http://dx.doi.org/10.2307/2007076. MR690845 · Zbl 0528.57008 · doi:10.2307/2007076.MR690845
[16] I. KIM, X. WAN, and G. ZHANG, Plurisuperharmonicity of reciprocal energy function on Te-ichmüller space and Weil-Petersson metric, J. Math. Pures Appl. (9) 141 (2020), 316-341, avail-able at https://arxiv.org/abs/arXiv:1901.05048 (English, with English and French sum-maries). http://dx.doi.org/10.1016/j.matpur.2020.01.009. MR4134458 · Zbl 0433.58012 · doi:10.14492/hokmj/1381758271
[17] , New Kähler metric on quasifuchsian space and its curvature properties, submitted.
[18] , Convexity of energy function associated to the harmonic maps between surfaces, submitted. · Zbl 0528.57008 · doi:10.2307/2007076
[19] G. SCHUMACHER, Asymptotics of Kähler-Einstein metrics on quasi-projective manifolds and an extension theorem on holomorphic maps, Math. Ann. 311 (1998), no. 4, 631-645. http://dx. doi.org/10.1007/s002080050203. MR1637968 · Zbl 0915.32002 · doi:10.1007/s002080050203.MR1637968
[20] , Positivity of relative canonical bundles and applications, Invent. Math. 190 (2012), no. 1, 1-56. http://dx.doi.org/10.1007/s00222-012-0374-7. MR2969273 · Zbl 1445.53049 · doi:10.1016/j.matpur.2020.01.009
[21] D. TOLEDO, Hermitian curvature and plurisubharmonicity of energy on Teichmüller space, Geom. Funct. Anal. 22 (2012), no. 4, 1015-1032. http://dx.doi.org/10.1007/s00039-012-0185-4. MR2984125 · Zbl 1254.32020 · doi:10.1007/s00039-012-0185-4.MR2984125
[22] A. J. TROMBA, Teichmüller Theory in Riemannian Geometry: Lecture Notes Prepared by Jochen Denzler, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. http://dx. doi.org/10.1007/978-3-0348-8613-0. MR1164870 · Zbl 0785.53001
[23] , Dirichlet’s energy on Teichmüller’s moduli space and the Nielsen realization problem, Math. Z. 222 (1996), no. 3, 451-464. http://dx.doi.org/10.1007/PL00004542. MR1400202 · Zbl 0849.32015 · doi:10.1007/PL00004542.MR1400202
[24] M. WOLF, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449-479. http://dx.doi.org/10.4310/jdg/1214442885. MR982185 · Zbl 0655.58009 · doi:10.4310/jdg/1214442885.MR982185
[25] , The Weil-Petersson Hessian of length on Teichmüller space, J. Differential Geom. 91 (2012), no. 1, 129-169. http://dx.doi.org/10.4310/jdg/1343133703. MR2944964 · Zbl 1254.30076 · doi:10.1007/978-3-0348-8613-0
[26] S. A. WOLPERT, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987), no. 2, 275-296. http://dx.doi.org/10.4310/jdg/1214440853. MR880186 · Zbl 0849.32015 · doi:10.1007/PL00004542
[27] , Convexity of geodesic-length functions: A reprise, Spaces of Kleinian Groups (Y. Minsky and M. Sakuma, eds.), London Math. Soc. Lecture Note Ser., vol. 329, Cambridge Univ. Press, Cambridge, 2006, pp. 233-245. MR2258752 · Zbl 0655.58009 · doi:10.4310/jdg/1214442885
[28] , Behavior of geodesic-length functions on Teichmüller space, J. Differential Geom. 79 (2008), no. 2, 277-334. http://dx.doi.org/10.4310/jdg/1211512642. MR2420020 · Zbl 1254.30076 · doi:10.4310/jdg/1343133703
[29] Y. XIN, Geometry of Harmonic Maps, Progress in Nonlinear Differential Equations and their Applications, vol. 23, Birkhäuser Boston, Inc., Boston, MA, 1996. http://dx.doi.org/10. 1007/978-1-4612-4084-6. MR1391729 · Zbl 0616.53039 · doi:10.4310/jdg/1214440853
[30] S. YAMADA, Weil-Petersson convexity of the energy functional on classical and universal Teichmüller spaces, J. Differential Geom. 51 (1999), no. 1, 35-96. http://dx.doi.org/10.4310/jdg/ 1214425025. MR1703604 · Zbl 1035.32009 · doi:10.4310/jdg/1214425025.MR1703604
[31] , Local and global aspects of Weil-Petersson geometry, Handbook of Teichmüller Theory.
[32] Vol. IV, IRMA Lect. Math. Theor. Phys., vol. 19, Eur. Math. Soc., Zürich, 2014, pp. 43-111. http://dx.doi.org/10.4171/117-1/2. MR3289699 · doi:10.1007/978-1-4612-4084-6
[33] INKANG KIM: School of Mathematics KIAS, Heogiro 85, Dongdaemun-gu Seoul, 130-722, Republic of Korea E-MAIL: inkang@kias.re.kr XUEYUAN WAN: Mathematical Science Research Center Chongqing University of Technology Chongquing 400054, China E-MAIL: xwan@cqut.edu.cn GENKAI ZHANG: Mathematical Sciences Chalmers University of Technology and Mathematical Sciences Göteborg University SE-41296 Göteborg, Sweden E-MAIL: genkai@chalmers.se KEY WORDS AND PHRASES: Harmonic map, Teichmüller space, energy function, Weil-Petersson metric. 2000 MATHEMATICS SUBJECT CLASSIFICATION: 53C43, 53C21, 53C25. Received: November 7, 2019. · Zbl 1035.32009 · doi:10.4310/jdg/1214425025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.