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ISSN 1088-6826(e) ISSN 0002-9939(p)

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

Author(s): 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 $K\le-1$ . If is a compactly supported function of bounded variation on , then 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 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 , if $c\colon[a,b]\to M$ is a closed curve and is the winding number of about , 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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