zbMATH — the first resource for mathematics

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.

45G05 Singular nonlinear integral equations
53A30 Conformal differential geometry (MSC2010)
Full Text: DOI EuDML arXiv
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.