## On global rigidity of the $p$-th root embedding

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- Proc. Amer. Math. Soc.
**144**(2016), 347-358 Request permission

## Abstract:

We study bona fide holomorphic isometric embeddings of the unit disk $\Delta$ into polydisks $\Delta ^p$ ($p\ge 2$) with sheeting number equal to $p$ and the assumption that all component functions of such embeddings are non-constant. We prove that all such embeddings are congruent to the $p$-th root embedding.## References

- Carlos A. Berenstein and Roger Gay,
*Complex variables*, Graduate Texts in Mathematics, vol. 125, Springer-Verlag, New York, 1991. An introduction. MR**1107514**, DOI 10.1007/978-1-4612-3024-3 - Otto Forster,
*Lectures on Riemann surfaces*, Graduate Texts in Mathematics, vol. 81, Springer-Verlag, New York-Berlin, 1981. Translated from the German by Bruce Gilligan. MR**648106** - Phillip Griffiths and Joseph Harris,
*Principles of algebraic geometry*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR**507725** - John B. Conway,
*Functions of one complex variable*, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR**503901** - Steven G. Krantz and Harold R. Parks,
*A primer of real analytic functions*, 2nd ed., BirkhĂ¤user Advanced Texts: Basler LehrbĂĽcher. [BirkhĂ¤user Advanced Texts: Basel Textbooks], BirkhĂ¤user Boston, Inc., Boston, MA, 2002. MR**1916029**, DOI 10.1007/978-0-8176-8134-0 - Ngaiming Mok,
*Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric*, J. Eur. Math. Soc. (JEMS)**14**(2012), no.Â 5, 1617â€“1656. MR**2966662**, DOI 10.4171/JEMS/343 - Ngaiming Mok,
*Geometry of holomorphic isometries and related maps between bounded domains*, Geometry and analysis. No. 2, Adv. Lect. Math. (ALM), vol. 18, Int. Press, Somerville, MA, 2011, pp.Â 225â€“270. MR**2882446** - Sui-Chung Ng,
*On holomorphic isometric embeddings of the unit disk into polydisks*, Proc. Amer. Math. Soc.**138**(2010), no.Â 8, 2907â€“2922. MR**2644903**, DOI 10.1090/S0002-9939-10-10305-0 - Sui-Chung Ng,
*On Holomorphic Isometric Embeddings from the Unit Disk into Polydisks and Their Generalizations*, Thesis at the University of Hong Kong, 2008. - Makoto Namba,
*Branched coverings and algebraic functions*, Pitman Research Notes in Mathematics Series, vol. 161, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR**933557**

## Additional Information

**Shan Tai Chan**- Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
- MR Author ID: 1132275
- Email: puremath.stschan@gmail.com
- Received by editor(s): August 13, 2014
- Received by editor(s) in revised form: October 29, 2014, November 3, 2014, November 6, 2014, and November 14, 2014
- Published electronically: June 9, 2015
- Communicated by: Lei Ni
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 347-358 - MSC (2010): Primary 53C35, 53C55
- DOI: https://doi.org/10.1090/proc/12674
- MathSciNet review: 3415601