The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces

Author:
Ralph Howard

Journal:
Proc. Amer. Math. Soc. **126** (1998), 2779-2787

MSC (1991):
Primary 53C42; Secondary 53A04, 53C65

MathSciNet review:
1451806

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Abstract: Let be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature . If is a compactly supported function of bounded variation on , then satisfies the Sobolev inequality

Conversely, letting be the characteristic function of a domain recovers the sharp form of the isoperimetric inequality for simply connected surfaces with . Therefore this is the Sobolev inequality ``equivalent'' to the isoperimetric inequality for this class of surfaces. This is a special case of a result that gives the equivalence of more general isoperimetric inequalities and Sobolev inequalities on surfaces.

Under the same assumptions on , if is a closed curve and is the winding number of about , then the Sobolev inequality implies

which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature .

**1.**Thomas F. Banchoff and William F. Pohl,*A generalization of the isoperimetric inequality*, J. Differential Geometry**6**(1971/72), 175–192. MR**0305319****2.**Yu. D. Burago and V. A. Zalgaller,*Geometric inequalities*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR**936419****3.**Isaac Chavel,*Eigenvalues in Riemannian geometry*, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR**768584****4.**Kazuyuki Enomoto,*A generalization of the isoperimetric inequality on 𝑆² and flat tori in 𝑆³*, Proc. Amer. Math. Soc.**120**(1994), no. 2, 553–558. MR**1163333**, 10.1090/S0002-9939-1994-1163333-7**5.**Herbert Federer and Wendell H. Fleming,*Normal and integral currents*, Ann. of Math. (2)**72**(1960), 458–520. MR**0123260****6.**Wendell H. Fleming and Raymond Rishel,*An integral formula for total gradient variation*, Arch. Math. (Basel)**11**(1960), 218–222. MR**0114892****7.**Liliana M. Gysin,*The isoperimetric inequality for nonsimple closed curves*, Proc. Amer. Math. Soc.**118**(1993), no. 1, 197–203. MR**1079698**, 10.1090/S0002-9939-1993-1079698-X**8.**Robert Osserman,*The isoperimetric inequality*, Bull. Amer. Math. Soc.**84**(1978), no. 6, 1182–1238. MR**0500557**, 10.1090/S0002-9904-1978-14553-4**9.**B. Süssmann,*Curve shorting and the Banchoff-Pohl inequality on surfaces of nonpositive curvature*, Preprint (1996).**10.**Eberhard Teufel,*A generalization of the isoperimetric inequality in the hyperbolic plane*, Arch. Math. (Basel)**57**(1991), no. 5, 508–513. MR**1129528**, 10.1007/BF01246751**11.**Eberhard Teufel,*Isoperimetric inequalities for closed curves in spaces of constant curvature*, Results Math.**22**(1992), no. 1-2, 622–630. MR**1174928**, 10.1007/BF03323109**12.**E. Teufel,*On integral geometry in Riemannian spaces*, Abh. Math. Sem. Univ. Hamburg**63**(1993), 17–27. MR**1227860**, 10.1007/BF02941328**13.**Joel L. Weiner,*A generalization of the isoperimetric inequality on the 2-sphere*, Indiana Univ. Math. J.**24**(1974/75), 243–248. MR**0380687****14.**Joel L. Weiner,*Isoperimetric inequalities for immersed closed spherical curves*, Proc. Amer. Math. Soc.**120**(1994), no. 2, 501–506. MR**1163337**, 10.1090/S0002-9939-1994-1163337-4**15.**Shing Tung Yau,*Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold*, Ann. Sci. École Norm. Sup. (4)**8**(1975), no. 4, 487–507. MR**0397619****16.**William P. Ziemer,*Weakly differentiable functions*, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR**1014685**

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Additional Information

**Ralph Howard**

Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Email:
howard@math.sc.edu

DOI:
https://doi.org/10.1090/S0002-9939-98-04336-6

Keywords:
Isoperimetric inequalities,
Sobolev inequalities,
Banchoff-Pohl inequality

Received by editor(s):
June 21, 1996

Received by editor(s) in revised form:
February 6, 1997

Communicated by:
Christopher Croke

Article copyright:
© Copyright 1998
American Mathematical Society