## Duke Mathematical Journal

### Isometries between the spaces of $L^1$ holomorphic quadratic differentials on Riemann surfaces of finite type

#### Abstract

By applying the methods of V. Markovic [7] to the special case of Riemann surfaces of finite type, we obtain a transparent new proof of a classical result about isometries between the spaces of $L^1$ holomorphic quadratic differentials on such surfaces.

#### Article information

Source
Duke Math. J., Volume 120, Number 2 (2003), 433-440.

Dates
First available in Project Euclid: 16 April 2004

https://projecteuclid.org/euclid.dmj/1082138591

Digital Object Identifier
doi:10.1215/S0012-7094-03-12029-3

Mathematical Reviews number (MathSciNet)
MR2019983

Zentralblatt MATH identifier
1063.30038

#### Citation

Earle, Clifford J.; Markovic, V. Isometries between the spaces of $L^1$ holomorphic quadratic differentials on Riemann surfaces of finite type. Duke Math. J. 120 (2003), no. 2, 433--440. doi:10.1215/S0012-7094-03-12029-3. https://projecteuclid.org/euclid.dmj/1082138591

#### References

• \lccS. Banach, Theory of Linear Operations, trans. F. Jellett, North-Holland Math. Library 38, North-Holland, Amsterdam, 1987.
• \lccC. J. Earle and I. Kra, "On holomorphic mappings between Teichmüller spaces" in Contributions to Analysis, Academic Press, New York, 1974, 107–124.
• ––––, On isometries between Teichmüller spaces, Duke Math. J. 41 (1974), 583–591.
• \lccO. Forster, Lectures on Riemann Surfaces, trans. B. Gilligan, Grad. Texts in Math. 81, Springer, New York, 1991.
• \lccN. Lakic, An isometry theorem for quadratic differentials on Riemann surfaces of finite genus, Trans. Amer. Math. Soc. 349 (1997), 2951–2967.
• \lccJ. Lamperti, On the isometries of certain function–spaces, Pacific J. Math. 8 (1958), 459–466.
• \lccV. Markovic, Biholomorphic maps between Teichmüller spaces, Duke Math. J. 120 (2003), 405–431.
• \lccS. Nag, The Complex Analytic Theory of Teichmüller Spaces, Canad. Math. Soc. Ser. Monogr. Adv. Texts, Wiley, New York, 1988.
• \lccH. L. Royden, "Automorphisms and isometries of Teichmüller space" in Advances in the Theory of Riemann Surfaces (Stony Brook, N.Y., 1969), Ann. of Math. Stud. 66, Princeton Univ. Press, Princeton, 1971, 369–383.
• ––––, Real Analysis, 3rd ed., Macmillan, New York, 1988.
• \lccW. Rudin, $L^{p}$-isometries and equimeasurability, Indiana Univ. Math. J. 25 (1976), 215–228.