Metric topology on the moduli space

Jialong Deng

https://orcid.org/0000-0002-2556-2259

Germany

Georg-August-Universität

Mathematisches Institut

Submitted: 2020-01-28

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Accepted: 2020-12-28

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Published: 2021-04-01

DOI: https://doi.org/10.4995/agt.2021.13066
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Keywords:

Gromov-Hausdorff topology, ε-topology, Lipschitz-topology, smooth Lipschitz-topology

Supporting agencies:

University of Goettingen

Abstract:

We define the smooth Lipschitz topology on the moduli space and show that each conformal class is dense in the moduli space endowed with Gromov-Hausdorff topology, which offers an answer to Tuschmann’s question.

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References:

N. A'Campo, L. Ji and A. Papadopoulos, On the early history of moduli and Teichmüller spaces, arXiv e-prints, page arXiv:1602.07208, Feb 2016.

D. Burago, Y. Burago and S. Ivanov, A course in metric geometry, volume 33, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. https://doi.org/10.1090/gsm/033

D. A. Edwards, The structure of superspace, in: Studies in topology (Proc. Conf., Univ. North Carolina, Charlotte, N. C., 1974 dedicated to Math. Sect. Polish Acad. Sci.), pages 121-133, 1975. https://doi.org/10.1016/B978-0-12-663450-1.50017-7

A. E. Fischer, The theory of superspace, in: Relativity (Proc. Conf. Midwest, Cincinnati, Ohio, 1969), pages 303-357, 1970. https://doi.org/10.1007/978-1-4684-0721-1_16

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53-73. https://doi.org/10.1007/BF02698687

M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA (2007).

C.-H. Li, Quantum fluctuations, conformal deformations, and Gromov's topology - Wheeler, DeWitt, and Wilson meeting Gromov, arXiv e-prints, page arXiv:1509.03895, Sep 2015.

W. Tuschmann, Spaces and moduli spaces of Riemannian metrics, Front. Math. China 11, no. 5 (2016), 1335-1343. https://doi.org/10.1007/s11464-016-0576-1

W. Tuschmann and D. J. Wraith, Moduli spaces of Riemannian metrics, volume 46, Oberwolfach Seminars, Birkhäuser Verlag, Basel, 2015. https://doi.org/10.1007/978-3-0348-0948-1

J. A. Wheeler, Superspace, in: Analytic methods in mathematical physics (Sympos., Indiana Univ., Bloomington, Ind., 1968), pages 335-378, 1970.

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