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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A Riemannian four vertex theorem for surfaces with boundary


Author: Mohammad Ghomi
Journal: Proc. Amer. Math. Soc. 139 (2011), 293-303
MSC (2010): Primary 53C20, 53C22; Secondary 53A04, 53A05
Published electronically: July 22, 2010
MathSciNet review: 2729091
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that every metric of constant curvature on a compact surface $ M$ with boundary $ \partial M$ induces at least four vertices, i.e., local extrema of geodesic curvature on a connected component of $ \partial M$ if, and only if, $ M$ is simply connected. Indeed, when $ M$ is not simply connected, we construct hyperbolic, parabolic, and elliptic metrics of constant curvature on $ M$ with only two critical points of geodesic curvature on each component of $ \partial M$. With few exceptions, these metrics are obtained by removing the singularities and a perturbation of flat structures on closed surfaces.


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Additional Information

Mohammad Ghomi
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: ghomi@math.gatech.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10507-5
PII: S 0002-9939(2010)10507-5
Keywords: Four vertex theorems, flat structures with conical singularities, surfaces of constant curvature with boundary, space forms, geodesic vertices.
Received by editor(s): September 30, 2009
Received by editor(s) in revised form: March 19, 2010
Published electronically: July 22, 2010
Additional Notes: The author was supported by NSF Grant DMS-0336455, and CAREER award DMS-0332333.
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.