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Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups


Authors: Luca Capogna, Scott D. Pauls and Jeremy T. Tyson
Journal: Trans. Amer. Math. Soc. 362 (2010), 4045-4062
MSC (2000): Primary 43A80, 53C17; Secondary 22E30, 35H20, 52A41, 53A35
DOI: https://doi.org/10.1090/S0002-9947-10-04768-9
Published electronically: March 12, 2010
MathSciNet review: 2608394
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Abstract | References | Similar Articles | Additional Information

Abstract: We use a Riemannian approximation scheme to give a characterization for smooth convex functions on a Carnot group (in the sense of Danielli-Garofalo-Nhieu or Lu-Manfredi-Stroffolini) in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph.


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Additional Information

Luca Capogna
Affiliation: Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701
Email: lcapogna@uark.edu

Scott D. Pauls
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
Email: scott.pauls@dartmouth.edu

Jeremy T. Tyson
Affiliation: Department of Mathematics, University of Illinois, Urbana-Champaign, Illinois 61801
Email: tyson@math.uiuc.edu

DOI: https://doi.org/10.1090/S0002-9947-10-04768-9
Keywords: Convexity, Carnot group, second fundamental form, Riemannian approximation
Received by editor(s): May 15, 2006
Received by editor(s) in revised form: November 14, 2007
Published electronically: March 12, 2010
Additional Notes: The authors were partially supported by the National Science Foundation: the first author was partially supported by NSF DMS-0134318; the second author was partially supported by NSF DMS-0306752; the third author was partially supported by NSF DMS-0228807, DMS-0555869; all authors were partially supported by NSF DMS-0503695. Part of the research for this paper was done while the first and third authors were visitors of the Mathematics Department at Dartmouth College in Spring 2006. They gratefully acknowledge the hospitality.
Article copyright: © Copyright 2010 American Mathematical Society
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