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Proceedings of the American Mathematical Society Series B
Proceedings of the American Mathematical Society Series B
ISSN 2330-1511

 

Height estimate for special Weingarten surfaces of elliptic type in $ {\mathbb{M}}^2(c) \times\mathbb{R}$


Author: Filippo Morabito
Journal: Proc. Amer. Math. Soc. Ser. B 1 (2014), 14-22
MSC (2010): Primary 53A10
Published electronically: January 10, 2014
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Abstract: In this article we provide a vertical height estimate for compact special Weingarten surfaces of elliptic type in $ {\mathbb{M}}^2(c) \times \mathbb{R}$, i.e. surfaces whose mean curvature $ H$ and extrinsic Gauss curvature $ K_e$ satisfy $ H=f(H^2-K_e)$ with $ 4x(f'(x))^2<1,$ for all $ x \in [0,+\infty ).$ The vertical height estimate generalizes a result by Rosenberg and Sa Earp and applies only to surfaces verifying a height estimate condition. When $ c<0,$ using also a horizontal height estimate, we show a non-existence result for properly embedded Weingarten surfaces of elliptic type in $ \mathbb{H}^2(c) \times \mathbb{R}$ with finite topology and one end.


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Additional Information

Filippo Morabito
Affiliation: Korea Institute for Advanced Study, Cheongnyangni 2-dong, Dongdaemun-gu, Seoul, 130-722, South Korea
Address at time of publication: Department of Mathematical Sciences, Korea Advanced Institute Science Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, South Korea

DOI: http://dx.doi.org/10.1090/S2330-1511-2014-00005-5
PII: S 2330-1511(2014)00005-5
Keywords: Special Weingarten surfaces, ellipticity, height estimate
Received by editor(s): July 31, 2011
Received by editor(s) in revised form: June 20, 2012, January 29, 2013, March 29, 2013, and April 3, 2013
Published electronically: January 10, 2014
Communicated by: Michael Wolf
Article copyright: © Copyright 2014 by the author under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)