# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfacesHTML articles powered by AMS MathViewer

by Ralph Howard
Proc. Amer. Math. Soc. 126 (1998), 2779-2787 Request permission

## 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 $4\pi \int _M f^2 dA+ \left (\int _M |f| dA \right )^2\le \left (\int _M\|\nabla f\| dA \right )^2.$ 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 $4\pi \int _M w_c^2 dA+ \left (\int _M|w_c| dA \right )^2\le L(c)^2,$ which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature $\le -1$.
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