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An integral equation in conformal geometry. (English) Zbl 1154.45004
Summary: Motivated by Carleman’s proof of the isoperimetric inequality in the plane, we study the problem of finding a metric with zero scalar curvature maximizing the isoperimetric ratio among all zero scalar curvature metrics in a fixed conformal class on a compact manifold with boundary. We derive a criterion for the existence and make a related conjecture.

MSC:
45G05 Singular nonlinear integral equations
53A30 Conformal differential geometry (MSC2010)
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[1] Adams, R.A., Sobolev spaces, Pure and applied mathematics, vol. 65, (2003), Academic Press New York-London · Zbl 0186.19101
[2] Baernstein, A.; Taylor, B.A., Spherical rearrangements, sub-harmonic functions and *-functions in n-space, Duke math. J., 43, 245-268, (1976) · Zbl 0331.31002
[3] Carleman, T., Zur theorie der minimalflächen, Math. Z., 9, 154-160, (1921) · JFM 48.0590.02
[4] Escobar, J.F., The Yamabe problem on manifolds with boundary, J. differential geom., 35, 1, 21-84, (1992) · Zbl 0771.53017
[5] Escobar, J.F., Conformal deformation of a riemannian metric to a scalar flat metric with constant Mean curvature on the boundary, Ann. of math. (2), 136, 1, 1-50, (1992) · Zbl 0766.53033
[6] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1998), Springer-Verlag Berlin · Zbl 0691.35001
[7] F.B. Hang, X.D. Wang, X.D. Yan, Sharp integral inequalities for harmonic functions, Comm. Pure Appl. Math., in press · Zbl 1173.26321
[8] S. Jacobs, An isoperimetric inequality for functions analytic in multiply connected domains, Mittag-Leffler Institute report, 1972
[9] Lee, J.M.; Parker, T.H., The Yamabe problem, Bull. amer. math. soc. (N.S.), 17, 1, 37-91, (1987) · Zbl 0633.53062
[10] Lions, P.L., The concentration-compactness principle in the calculus of variations. the limit case. II, Rev. mat. iberoamericana, 1, 2, 45-121, (1985) · Zbl 0704.49006
[11] Stein, E.M.; Weiss, G., Introduction to Fourier analysis on Euclidean spaces, Princeton mathematical series, vol. 32, (1971), Princeton University Press Princeton, NJ · Zbl 0232.42007
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