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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 $(M,g)$ be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature $K\le-1$. If $f$ is a compactly supported function of bounded variation on $M$, then $f$ satisfies the Sobolev inequality

\begin{displaymath}4\pi \int _M f^2\,dA+ \left(\int _M |f|\,dA \right)^2\le \left(\int _M\|\nabla f\|\,dA \right)^2. \end{displaymath}

Conversely, letting $f$ be the characteristic function of a domain $D\subset M$ recovers the sharp form $4\pi A(D)+A(D)^2\le L(\partial D)^2$ of the isoperimetric inequality for simply connected surfaces with $K\le -1$. 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 $(M,g)$, if $c\colon[a,b]\to M$ is a closed curve and $w_c(x)$ is the winding number of $c$ about $x$, then the Sobolev inequality implies

\begin{displaymath}4\pi\int _M w_c^2\,dA+ \left(\int _M|w_c|\,dA \right)^2\le L(c)^2, \end{displaymath}

which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature $\le -1$.

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Additional Information

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

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