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

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- by Ralph Howard PDF
- 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 â€śequivalentâ€ť 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), 175â€“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. SosinskiÄ; 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.Â 2, 553â€“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), 458â€“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), 218â€“222. MR**114892**, DOI 10.1007/BF01236935 - Liliana M. Gysin,
*The isoperimetric inequality for nonsimple closed curves*, Proc. Amer. Math. Soc.**118**(1993), no.Â 1, 197â€“203. MR**1079698**, DOI 10.1090/S0002-9939-1993-1079698-X - Robert Osserman,
*The isoperimetric inequality*, Bull. Amer. Math. Soc.**84**(1978), no.Â 6, 1182â€“1238. MR**500557**, DOI 10.1090/S0002-9904-1978-14553-4 - B. SĂĽssmann,
*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.Â 5, 508â€“513. MR**1129528**, DOI 10.1007/BF01246751 - Eberhard Teufel,
*Isoperimetric inequalities for closed curves in spaces of constant curvature*, Results Math.**22**(1992), no.Â 1-2, 622â€“630. MR**1174928**, DOI 10.1007/BF03323109 - E. Teufel,
*On integral geometry in Riemannian spaces*, Abh. Math. Sem. Univ. Hamburg**63**(1993), 17â€“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), 243â€“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.Â 2, 501â€“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. Ă‰cole Norm. Sup. (4)**8**(1975), no.Â 4, 487â€“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

## Additional Information

**Ralph Howard**- Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
- MR Author ID: 88825
- Email: howard@math.sc.edu
- Received by editor(s): June 21, 1996
- Received by editor(s) in revised form: February 6, 1997
- Communicated by: Christopher Croke
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**126**(1998), 2779-2787 - MSC (1991): Primary 53C42; Secondary 53A04, 53C65
- DOI: https://doi.org/10.1090/S0002-9939-98-04336-6
- MathSciNet review: 1451806