Abstract
Over the space of Kähler metrics associated to a fixed Kähler class, we first prove the lower bound of the energy functional \(\tilde{E}^\beta \), then we provide the criteria of the geodesics rays to detect the lower bound of \(\tilde{{\mathfrak {J}}}^\beta \)-functional . They are used to obtain the I-properness of Mabuchi’s K-energy. The criteria are examined under a necessary and sufficient condition by showing the convergence of the negative gradient flow of \(\tilde{{\mathfrak {J}}}^\beta \)-functional.
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References
Aubin, T.: Some nonlinear problems in Riemannian geometry. In: Springer Monographs in Mathematics. Springer, Berlin (1998)
Calabi, E., Chen, X.: The space of Kähler metrics. II. J. Differ. Geom. 61(2), 173–193 (2002)
Calamai, S., Zheng, K.: Geodesics in the space of Kähler cone metrics, I. To appear in Am. J. Math. (2012) arXiv:1205.0056v4
Chen, X.: A new parabolic flow in Kähler manifolds. Commun. Anal. Geom. 12(4), 837–852 (2004)
Chen, X.: On the lower bound of the Mabuchi energy and its application. Int. Math. Res. Not. 12, 607–623 (2000)
Chen, X.: Space of Kähler metrics. IV. On the lower bound of the \(K\)-energy (2008). arXiv:0809.4081v2
Chen, X.: Space of Kähler metrics. III. On the lower bound of the Calabi energy and geodesic distance. Invent. Math. 175(3), 453–503 (2009)
Chen, X.: The space of Kähler metrics. J. Differ. Geom. 56(2), 189–234 (2000)
Ding, W.Y.: Remarks on the existence problem of positive Kähler–Einstein metrics. Math. Ann. 282(3), 463–471 (1988)
Ding, W.Y., Tian, G.: The generalized Moser–Trudinger inequality. In: Proceedings of Nankai International Conference on Nonlinear Analysis (1993)
Donaldson, S.K.: Moment maps and diffeomorphisms. Asian J. Math. 3(1), 1–15 (1999). (Sir Michael Atiyah: a great mathematician of the twentieth century)
Donaldson, S.K.: Symmetric spaces, Kähler geometry and Hamiltonian dynamics. In: Northern California Symplectic Geometry Seminar. Am. Math. Soc. Transl. Ser. 2, vol. 196, pp. 13–33. Am. Math. Soc., Providence (1999)
Fang, H., Lai, M., Ma, X.: On a class of fully nonlinear flows in Kähler geometry. J. Reine Angew. Math. 653, 189–220 (2011)
Fang, H., Lai, M., Song, J., Weinkove, B.: The \(J\)-flow on Kähler surfaces: a boundary case. Anal. PDE 7(1), 215–226 (2014)
Guan, B., Li, Q.: The Dirichlet problem for a complex Monge–Ampère type equation on Hermitian manifolds. Adv. Math. 246, 351–367 (2013)
Lejmi, M., Székelyhidi, G.: The \(J\)-flow and stability. Adv. Math. 274, 404–431 (2015)
Li, H., Shi, Y., Yao, Y.: A criterion for the properness of the \(K\)-energy in a general Kähler class. Math. Ann. 361(1–2), 135–156 (2015)
Li, H., Zheng, K.: Kähler non-collapsing, eigenvalues and the Calabi flow. J. Funct. Anal. 267(5), 1593–1636 (2014)
Mabuchi, T.: \(K\)-energy maps integrating Futaki invariants. Tohoku Math. J. (2) 38(4), 575–593 (1986)
Moser, J.K.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970/71)
Onofri, E.: On the positivity of the effective action in a theory of random surfaces. Commun. Math. Phys. 86(3), 321–326 (1982)
Phong, D.H., Song, J., Sturm, J., Weinkove, B.: The Moser–Trudinger inequality on Kähler–Einstein manifolds. Am. J. Math. 130(4), 1067–1085 (2008)
Semmes, S.: Complex Monge–Ampère and symplectic manifolds. Am. J. Math. 114(3), 495–550 (1992)
Song, J., Weinkove, B.: On the convergence and singularities of the \(J\)-flow with applications to the Mabuchi energy. Commun. Pure Appl. Math. 61(2), 210–229 (2008)
Tian, G.: On Kähler–Einstein metrics on certain Kähler manifolds with \(c_1(M)>0\). Invent. Math. 89(2), 225–246 (1987)
Tian, G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math. 130(1), 1–37 (1997)
Tian, G.: Canonical metrics in Kähler geometry. In: Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2000). (Notes taken by Meike Akveld )
Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)
Weinkove, B.: Convergence of the \(J\)-flow on Kähler surfaces. Commun. Anal. Geom. 12(4), 949–965 (2004)
Weinkove, B.: On the \(J\)-flow in higher dimensions and the lower boundedness of the Mabuchi energy. J. Differ. Geom. 73(2), 351–358 (2006)
Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I. Commun. Pure Appl. Math. 31(3), 339–411 (1978)
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Communicated by J. Jost.
