The isoperimetric problem on surfaces

American Mathematical Monthly

Volumn 106, Issue 5, 1999, Pages 430-439

  Howards, Hugh ^a   Hutchings, Michael ^b   Morgan, Frank ^c  

^a WAKE FOREST UNIVERSITY   (United States)

^b STANFORD UNIVERSITY   (United States)

^c WILLIAMS COLLEGE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033449295     PISSN: 00029890     EISSN: None     Source Type: Journal    
DOI: 10.2307/2589147     Document Type: Article

References (30)

1. Colin Adams and Frank Morgan, Isoperimetric curves on hyperbolic surfaces, Proc. Amer. Math. Soc., to appear.
2. Itai Benjamini and Jianguo Cao, A new isoperimetric theorem for surfaces of variable curvature, Duke Math. J. 85 (1996) 359-396.
3. M. Berger, Geometry II, Springer-Verlag, New York, 1977.
4. W. Blaschke, Vorlesungen über Differentialgeometrie II: Affine Differentialgeometrie, Berlin, 1923.
5. T. Bonnesen and W. Fenchel, Theory of Convex Bodies (English translation of Theorie der Konvexen Körper), BCS Associates, Moscow, Idaho, 1987.
6. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Wiley & Sons, NY, 1953.
7. C. Croke, Poincare's problem on the shortest closed geodesic on a convex hypersurface, J. Differential Geom. 17 (1982) 595-634.
8. Harold Gordon Eggleston, Convexity, Cambridge University Press, Cambridge, 1966.
9. 3, undergraduate thesis, Williams College, 1991.
10. M. Gromov, Isoperimetric inequalities in Riemannian manifolds, Appendix I to Vitali D. Milman and Gideon Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Mathematics, No. 1200, Springer-Verlag, New York, 1986.
11. Joel Hass, Michael Hutchings, and Roger Schlafly, The double bubble conjecture, Electron. Res. Announc. Amer. Math. Soc. 1 (1995) 98-102.
12. Joel Hass and Frank Morgan, Geodesics and soap bubbles in surfaces, Math. Z. 223 (1996) 185-196.
13. Joel Hass and Roger Schlafly, Bubbles and double bubbles, American Scientist, Sept.-Oct., 1996, pp. 462-467.
14. Joel Hass and Roger Schlafly, Double bubbles minimize, Ann. of Math., to appear.
15. Hugh Howards, Soap bubbles on surfaces, undergraduate thesis, Williams College, 1992.
16. Michael Hutchings, The structure of area-minimizing double bubbles, J. Geom. Anal. 7 (1997) 285-304.
17. Frank Morgan, Clusters minimizing area plus length of singular curves, Math. Ann. 299 (1994) 697-714.
18. Frank Morgan, The Double Bubble Conjecture, FOCUS, Math. Assn. Amer., December, 1995.
19. Frank Morgan, Geometric Measure Theory: a Beginner's Guide, Academic Press, second edition, Boston, 1995.
20. Frank Morgan, Riemannian Geometry: a Beginner's Guide, A. K. Peters, Ltd., second edition, Natick, 1998.
21. 2 and in surfaces, Pacific J. Math. 165 (1994) 347-361.
22. Frank Morgan, Michael Hutchings, and Hugh Howards, The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature, Trans. Amer. Math. Soc., to appear.
23. Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978) 1182-1238.
24. Pierre Pansu, Sur la régularité du profil isopérimétrique des surfaces riemanniennes compactes, Ann. Inst. Fourier 48 (1998) 247-264.
25. Thomas Isaac Porter, A history of the classical isoperimetric problem, in Contributions to the Calculus of Variations 1931-1932, Univ. of Chicago Press, Chicago, 1933, pp. 475-520.
26. Manuel Ritoré, Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces, preprint (1998).
27. J. Steiner, Einfache Beweise der isoperimetrischen Hauptsatze, Crelle's J. 18 (1838) 286-287.
28. J. Steiner, Sur le maximum et minimum des figures dans le plan, sur le sphere, et l'espace en general, Crelle's J. 24 (1842) 93-152, 189-250 (figures separate at end of fascicle).
29. Peter Topping, The isoperimetric inequality on a surface, preprint (1997).
30. Peter Topping, Mean curvature flow and geometric inequalities, J. Reine Angew. Math. 503 (1998) 47-61.