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The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces
Authors:Ralph Howard
Affiliation:Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Abstract:
Let $(M,g)$ be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature $Kle-1$. If $f$ is a compactly supported function of bounded variation on $M$, then $f$ satisfies the Sobolev inequality

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

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

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

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

Keywords:Isoperimetric inequalities   Sobolev inequalities   Banchoff-Pohl inequality
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