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

Howard, Ralph

doi:10.1090/S0002-9939-98-04336-6

The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces
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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 $\le -1$.

References

Thomas F. Banchoff and William F. Pohl, A generalization of the isoperimetric inequality, J. Differential Geometry 6 (1971/72), 175–192. MR 305319
Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419, DOI 10.1007/978-3-662-07441-1
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Kazuyuki Enomoto, A generalization of the isoperimetric inequality on $S^2$ and flat tori in $S^3$, Proc. Amer. Math. Soc. 120 (1994), no. 2, 553–558. MR 1163333, DOI 10.1090/S0002-9939-1994-1163333-7
Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
Wendell H. Fleming and Raymond Rishel, An integral formula for total gradient variation, Arch. Math. (Basel) 11 (1960), 218–222. MR 114892, DOI 10.1007/BF01236935
Liliana M. Gysin, The isoperimetric inequality for nonsimple closed curves, Proc. Amer. Math. Soc. 118 (1993), no. 1, 197–203. MR 1079698, DOI 10.1090/S0002-9939-1993-1079698-X
Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 500557, DOI 10.1090/S0002-9904-1978-14553-4
B. Süssmann, Curve shorting and the Banchoff-Pohl inequality on surfaces of nonpositive curvature, Preprint (1996).
Eberhard Teufel, A generalization of the isoperimetric inequality in the hyperbolic plane, Arch. Math. (Basel) 57 (1991), no. 5, 508–513. MR 1129528, DOI 10.1007/BF01246751
Eberhard Teufel, Isoperimetric inequalities for closed curves in spaces of constant curvature, Results Math. 22 (1992), no. 1-2, 622–630. MR 1174928, DOI 10.1007/BF03323109
E. Teufel, On integral geometry in Riemannian spaces, Abh. Math. Sem. Univ. Hamburg 63 (1993), 17–27. MR 1227860, DOI 10.1007/BF02941328
Joel L. Weiner, A generalization of the isoperimetric inequality on the $2$-sphere, Indiana Univ. Math. J. 24 (1974/75), 243–248. MR 380687, DOI 10.1512/iumj.1974.24.24021
Joel L. Weiner, Isoperimetric inequalities for immersed closed spherical curves, Proc. Amer. Math. Soc. 120 (1994), no. 2, 501–506. MR 1163337, DOI 10.1090/S0002-9939-1994-1163337-4
Shing Tung Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 4, 487–507. MR 397619, DOI 10.24033/asens.1299
William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3