Variational structure of the \(v_{\frac{n}{2}}\)-Yamabe problem. (English) Zbl 1383.53028

Summary: We define a formal Riemannian metric on a conformal class in the context of the \(v_{\frac{n}{2}}\)-Yamabe problem. We also give a new variational description of this problem, and show that the associated functional is geodesically convex. Formal properties of the negative gradient flow are also described. These results parallel our work in two dimensions on the Liouville energy and the uniformization of surfaces, and our work in four dimensions on the \(\sigma_2\)-Yamabe problem in two preprints.


53C20 Global Riemannian geometry, including pinching
58E11 Critical metrics
Full Text: DOI arXiv


[1] Lin, M.; Trudinger, N. S., On some inequalities for elementary symmetric functions, Bull. Aust. Math. Soc., 50, 2, 317-326, (1994) · Zbl 0855.26006
[2] B. Andrews, unpublished.
[3] Branson, T.; Gilkey, P.; Pohjanpelto, J., Invariants of locally conformally flat manifolds, Trans. Am. Math. Soc., 347, 3, 939-953, (1995) · Zbl 0820.53022
[4] Brendle, S.; Viaclovsky, J., A variational characterization for \(\sigma_{\frac{n}{2}}\), Calc. Var., 20, 399-402, (2004) · Zbl 1059.53033
[5] Calabi, E.; Chen, X. X., The space of Kähler metrics II, J. Differ. Geom., 61, 173-193, (2002) · Zbl 1067.58010
[6] Chang, S. Y.A.; Fang, H., A class of variational functionals in conformal geometry, Int. Math. Res. Not., (2008)
[7] Chang, S. Y.A.; Yang, P., The inequality of Moser and Trudinger and applications to conformal geometry, dedicated to the memory of Jürgen K. Moser, Commun. Pure Appl. Math., 56, 8, 1135-1150, (2003) · Zbl 1049.53025
[8] Chow, B.; Lu, P.; Ni, L., Hamilton’s Ricci flow, Lectures in Contemporary Mathematics, (2006), Science Press Beijing New York · Zbl 1118.53001
[9] Donaldson, S. K., Symmetric spaces, Kähler geometry and Hamiltonian dynamics, (Northern California Symplectic Geometry Seminar, Am. Math. Soc. Transl. Ser. 2, vol. 196, (1999), Amer. Math. Soc. Providence), 13-33 · Zbl 0972.53025
[10] Graham, C. R., Volume and area renormalizations for conformally compact Einstein metrics, The Proceedings of the 19th Winter School “Geometry and Physics”, Srni, 1999, Rend. Circ. Mat. Palermo (2) Suppl., 63, 31-42, (2000) · Zbl 0984.53020
[11] Graham, C. R., Extended obstruction tensors and renormalized volume coefficients, Adv. Math., 220, 1956-1985, (2009) · Zbl 1161.53062
[12] Graham, C. R.; Juhl, A., Holographic formula for Q-curvature, Adv. Math., 216, 841-853, (2007) · Zbl 1147.53030
[13] Guan, P.; Viaclovsky, J. A.; Wang, G., Some properties of the Schouten tensor and applications to conformal geometry, Trans. Am. Math. Soc., 355, 3, 925-933, (2003) · Zbl 1022.53035
[14] M.J. Gursky, J. Streets, A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow, preprint.
[15] M.J. Gursky, J. Streets, A formal Riemannian structure on conformal classes and uniqueness for the \(\sigma_k\)-Yamabe problem, preprint. · Zbl 1383.53028
[16] Mabuchi, T., Some symplectic geometry on compact Kähler manifolds (1), Osaka J. Math., 24, 227-252, (1987) · Zbl 0645.53038
[17] Petrov, F.
[18] Semmes, S., Complex Monge-ampere equations and symplectic manifolds, Am. J. Math., 114, 495-550, (1992) · Zbl 0790.32017
[19] Viaclovsky, J. A., Conformally invariant Monge-Ampère equations: global solutions, Trans. Am. Math. Soc., 352, 9, 4371-4379, (2000) · Zbl 0951.35044
[20] Viaclovsky, J. A., Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J., 101, 2, 283-316, (2000) · Zbl 0990.53035
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