## Osaka Journal of Mathematics

### On the lower bound of the $K$-energy and $F$-functional

Haozhao Li

#### Abstract

Using Perelman's results on Kähler-Ricci flow, we prove that the $K$-energy is bounded from below if and only if the $F$-functional is bounded from below in the canonical Kähler class.

#### Article information

Source
Osaka J. Math., Volume 45, Number 1 (2008), 253-264.

Dates
First available in Project Euclid: 14 March 2008

https://projecteuclid.org/euclid.ojm/1205503567

Mathematical Reviews number (MathSciNet)
MR2416659

Zentralblatt MATH identifier
1138.53056

#### Citation

Li, Haozhao. On the lower bound of the $K$-energy and $F$-functional. Osaka J. Math. 45 (2008), no. 1, 253--264. https://projecteuclid.org/euclid.ojm/1205503567

#### References

• S. Bando and T. Mabuchi: Uniqueness of Einstein Kähler metrics modulo connected group actions; in Algebraic Geometry, Sendai, Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1985, 11--40.
• X.X. Chen: On the lower bound of the Mabuchi energy and its application, Internat. Math. Res. Notices (2000), 607--623.
• X.X. Chen: On the lower bound of energy functional $E_{1}$, I, A stability theorem on the Kähler-Ricci flow, J. Geom. Anal. 16 (2006), 23--38.
• X.X. Chen, H. Li and B. Wang: On the Kähler-Ricci flow with small initial $E_{1}$ energy (I)
• X.X. Chen and G. Tian: Ricci flow on Kähler-Einstein surfaces, Invent. Math. 147 (2002), 487--544.
• X.X. Chen and G. Tian: Ricci flow on Kähler-Einstein manifolds, Duke Math. J. 131 (2006), 17--73.
• W.Y. Ding and G. Tian: The generalized Moser-Trudinger inequality; in Proceedings of Nankai International Conference of Nonlinear Analysis, 1993.
• A. Futaki: Kähler-Einstein Metrics and Integral Invariants, Lecture Notes in Math. 1314, Springer, Berlin, 1988.
• H. Li: A new formula for the energy functionals $E_k$ and its applications
• T. Mabuchi: $K$-energy maps integrating Futaki invariants, Tohoku Math. J. (2) 38 (1986), 575--593.
• N. Sesum and G. Tian: Bounding scalar curvature and diameter along the Kähler-Ricci flow (after Perelman) and some applications, preprint.
• N. Pali: A consequence of a lower bound of the $K$-energy, Int. Math. Res. Not. (2005), 3081--3090.
• G. Perelman: Unpublished work on Kähler-Ricci flow.
• Y. Rubinstein: On energy functionals and the existence of Kähler-Einstein metric.
• G. Tian: Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), 1--37.
• G. Tian: Canonical Metrics in Kähler Geometry, Notes taken by Meike Akveld, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 2000.
• G. Tian and X. Zhu: Convergence of Kähler-Ricci flow, J. Amer. Math. Soc. 20 (2007), 675--699.