The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Ralph Howard

Institution:	Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

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 $c\colona,b]\to M$ 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 $\le -1$ .

Keywords:	Isoperimetric inequalities Sobolev inequalities Banchoff-Pohl inequality

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