The sharp Sobolev inequality and the BanchoffPohl inequality on surfaces
Author:
Ralph Howard
Journal:
Proc. Amer. Math. Soc. 126 (1998), 27792787
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 BanchoffPohl 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:
http://dx.doi.org/10.1090/S0002993998043366
PII:
S 00029939(98)043366
Keywords:
Isoperimetric inequalities,
Sobolev inequalities,
BanchoffPohl 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
