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Bulletin of the London Mathematical Society Advance Access published online on May 22, 2009
Bulletin of the London Mathematical Society, doi:10.1112/blms/bdp044
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© 2009 London Mathematical Society

Area of small disks

Christopher B. Croke

Department of Mathematics
University of Pennsylvania
Philadelphia, PA 19104-6395
USA

Received 7 October 2008. Revision received 18 February 2009.

This paper considers Riemannian metrics on 2-dimensional disks where all geodesics are minimizing. A sharp reverse isoperimetric inequality is proved. This in turn yields near optimal bounds for the area of disks as well as near optimal upper bounds on the first non-zero Neumann eigenvalue of the Laplacian in terms only of the radius.

2000 Mathematics Subject Classification 53C22, 53C24, 53C65, 53A99, 53C20.

Research supported by NSF grant DMS 07-04145 and the Max-Planck Institute, Bonn.


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