Abstract
(1.1) Let M be a two-dimensional C2-manifold endowed with a C2-Riemannian metric. We say that M is a generalized surface if the metric in M is allowed to degenerate at isolated points; such points are called singularities of the metric. In this paper we use the method of Fiala-Bol (cf. [12, 9]) to give a proof of the following general isoperimetric inequality.
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References
Alexandroff, A.D.: lsoperimetric inequalities for curved surfaces. Dokl. Akad. Nauk. SSSR 47,235-238 (1945)
Alexandroff, A.D.: Die innere Geometrie der Konvexen Flächen. Berlin: Akademie-Verlag 1955
Alexandroff, A.D., Streltsov, V.V.: Estimates of the length of a curve on a surface [Russian]. Dokl. Akad. Nauk. SSSR 93, 221-224 (1953)
Alexandroff, A.D., Streltsov, V.V.: lsoperimetric problem and estimates of the length of a curve on a surface. Two-dimensional manifolds of bounded curvature. Proc. Steklov lnst. Math. 76, 81–99 (1965)
Alexandroff, A.D., Zalgaller, V.A.: Intrinsic Geometry of Surfaces. Translations of Mathematical Monographs 15. Providence, R.I.: Amer. Math. Soc. 1967
Bandle, C.: On a differential inequality and its applications to geometry. Math. Z. 147, 253–261 (1976)
Beckenbach, E.F., Radó, T.: Subharmonic functions and surfaces of negative curvature. Trans. Amer. Math. Soc. 35, 662–674 (1933)
Bernstein, F.: ÜOber die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene. Math. Ann. 60, 117–136 (1905)
Bol, G.: Isoperimetrische Ungleichung für Bereiche auf Flächen. Jber. Deutsch. Math.-Verein. 51, 219-257 (1941)
Burago, Yu.D.: Note on the isoperimetric inequality on two-dimensional surfaces. Siberian Math. J. 14, 666–668 (1973)
Carleman, T.: Zur Theorie der Minimal-Flachen. Math. Z. 9, 154–160 (1921)
Fiala, F.: Le problème des isoperimètres sur les surfaces ouverts à courbure positive. Comment. Math. Helv. 13, 293–396 (1940-41)
Gromov, M.L., Rokhlin, V.A.: Embeddings and immersions in Riemannian Geometry. Russian Math. Surveys 25, 5, 1–57 (1970)
Huber, A.: On the isoperimetric inequality on surfaces of variable Gaussian curvature. Ann. of Math 60, 237–247 (1954)
Huber, A.: Zum potentialtheoretischen Aspekt der Alexandrowschen Flachentheorie. Comment. Math. Helv. 34, 99–126 (1960)
lonin, V.K.: On isoperimetric and various other inequalities for a manifold of bounded curvature. Siberian Math. J. 10, 329-342 (1969)
Osserman, R.: Bonessen-style isoperimetric inequalities. Preprint
Reschetniak, !.G.: Isothermal coordinates in manifolds of bounded curvature. Dokl. Akad. Nauk. SSSR 94, 631–633 (1954)
Schmidt, E.: Beweis der Isoperimetrischen Eigenschaft der Kugel im Hyperbo1ischen und Spharischen Raum jeder Dimensionenzahl. Math. Z. 46, 204-230 (1940)
Toponogov, V.A.: An isoperimetric inequality for surfaces whose Gaussian curvature is bounded above. Siberian Math. J. 10, 144-157 (1969)
Whitney, H.: Differentiable manifolds. Ann. of Math. 37, 645–680 (1936)
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Barbosa, J.L., Carmo, M.d. (2012). A Proof of a General Isoperimetric Inequality for Surfaces. In: Tenenblat, K. (eds) Manfredo P. do Carmo – Selected Papers. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25588-5_13
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