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The isoperimetric problem on some singular surfaces

Part of: Classical differential geometry Manifolds

Published online by Cambridge University Press:  09 April 2009

Andrew Cotton ,
David Freeman ,
Andrei Gnepp ,
Ting Ng ,
John Spivack  and
Cara Yoder
Andrew Cotton
Affiliation:
C/o Frank Morgan Department of Mathematics Williams CollegeWilliamstown, MA 01267USA e-mail: [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
David Freeman
Affiliation:
C/o Frank Morgan Department of Mathematics Williams CollegeWilliamstown, MA 01267USA e-mail: [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
Andrei Gnepp
Affiliation:
C/o Frank Morgan Department of Mathematics Williams CollegeWilliamstown, MA 01267USA e-mail: [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
Ting Ng
Affiliation:
C/o Frank Morgan Department of Mathematics Williams CollegeWilliamstown, MA 01267USA e-mail: [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
John Spivack
Affiliation:
C/o Frank Morgan Department of Mathematics Williams CollegeWilliamstown, MA 01267USA e-mail: [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
Cara Yoder
Affiliation:
C/o Frank Morgan Department of Mathematics Williams CollegeWilliamstown, MA 01267USA e-mail: [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
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Abstract

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We characterize least-perimeter enclosures of prescribed area on some piecewise smooth manifolds, including certain polyhedra, double spherical caps, and cylindrical cans.

MSC classification

Secondary: 49Q10: Optimization of shapes other than minimal surfaces 53A10: Minimal surfaces, surfaces with prescribed mean curvature
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 78 , Issue 2 , April 2005 , pp. 167 - 197
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Almgren, F., ‘Spherical symmetrization’, in: Proceedings of the International Workshop on Integral Functionals in Calculus of Variations, (Trieste, 1985) Suppl. Rend. Circ. Mat. Palermo (2) 15 (1987) pp. 11–25.Google Scholar
[2]Barber, M., Tice, J. and Wecht, B., ‘Geodesics and geodesic nets on regular polyhedra’, Williams College NSF ‘SMALL’ undergraduate research Geomtry Group report(1995).Google Scholar
[3]Bray, H. and Morgan, F., ‘An isoperimetric comparison theorem for Schwarzschild space and other manifolds’, Proc. Amer. Math. Soc. 130 (2002), 1467–1472.CrossRefGoogle Scholar
[4]Burago, Yu. D. and Zalgaller, V. A., Geometric inequalities (Springer, New York, 1980).Google Scholar
[5]Erchak, A., Melinck, T. and Nicholson, R., ‘Geodesic nets on regular polyhedra’, Williams College NSF ‘SMALL’, undergraduate research Geometry Group report (1996).Google Scholar
[6]Federer, H., Geometric measure theory (Springer, New York, 1969).Google Scholar
[7]Gnepp, A., Ng, T. and Yoder, C., ‘Isoperimetric domains on polyhedra and singular surfaces’, Williams College NSF ‘SMALL’ undergraduate research Geometry Group report (1998).Google Scholar
[8]Heppes, A., e-mail communication to M. Barber, J. Tice, B. Wecht and F. Morgan, 1995.Google Scholar
[9]Howards, H., Hutchings, M. and Morgan, F., ‘The isoperimetric problem on surfaces’, Amer. Math. Monthly 106 (1999), 430–439.CrossRefGoogle Scholar
[10]Ivanov, A. O. and Tuzhilin, A. A., Minimal networks: the Steiner problem, and its generalizations (CRC Press, Boca Raton, 1994).Google Scholar
[11]Morgan, F., ‘An isoperimetric inequality for the thread problem’, Bull. Austral. Math. Soc. 55 (1997), 489–495.CrossRefGoogle Scholar
[12]Morgan, F., Geometric measure theory: a beginner's guide, 3rd edition (Academic Press, San Diego, 2000).CrossRefGoogle Scholar
[13]Morgan, F., ‘Area-minimizing surfaces in cones’, Comm. Anal. Geom. 10 (2002), 971–983.CrossRefGoogle Scholar
[14]Morgan, F., ‘Regularity of isoperimetric hypersurfaces in Riemannian manifolds’, Trans. Amer. Math. Soc. 355 (2003), 5041–5052.CrossRefGoogle Scholar
[15]Morgan, F., ‘In polytopes, small balls about some vertex minimize perimeter’, J. Differential Geom., to appear.Google Scholar
[16]Morgan, F. and Ritoré, M., ‘Isoperimetric regions in cones’, Trans. Amer. Math. Soc. 354 (2002), 2327–2339.CrossRefGoogle Scholar