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

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

MathSciNet review:
1451806

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Abstract | References | Similar Articles | Additional Information

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 .

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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