The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature

Morgan, Frank; Hutchings, Michael; Howards, Hugh

doi:10.1090/S0002-9947-00-02482-X

The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature
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by Frank Morgan, Michael Hutchings and Hugh Howards PDF

Trans. Amer. Math. Soc. 352 (2000), 4889-4909 Request permission

Abstract:

We prove that the least-perimeter way to enclose prescribed area in the plane with smooth, rotationally symmetric, complete metric of nonincreasing Gauss curvature consists of one or two circles, bounding a disc, the complement of a disc, or an annulus. We also provide a new isoperimetric inequality in general surfaces with boundary.

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