Abstract
You [Y2] has conjectured that a Fano manifold admits a Kähler-Einstein metric if and only if it is stable in the sense of geometric invariant theory. There has been much progress on this conjecture by Tian [T1], [T2], Donaldson [Do1], [Do2], and others. The Mabuchi energy functional plays a central role in these ideas. We study the Ek functionals introduced by X. X. Chen and G. Tian [CT1] which generalize the Mabuchi energy. We show that if a Fano manifold admits a Kähler-Einstein metric, then the functional E1 is bounded from below and, modulo holomorphic vector fields, is proper. This answers affirmatively a question raised by Chen [C2]. In fact, we show that E1 is proper if and only if there exists a Kähler-Einstein metric, giving a new analytic criterion for the existence of this canonical metric, with possible implications for the study of stability. We also show that on a Fano Kähler-Einstein manifold, all of the functionals Ek are bounded below on the space of metrics with nonnegative Ricci curvature.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 159-184 |
| Number of pages | 26 |
| Journal | Duke Mathematical Journal |
| Volume | 137 |
| Issue number | 1 |
| DOIs | |
| State | Published - Mar 15 2007 |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
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Energy functionals and canonical Kähler metrics. / Song, Jian; Weinkove, Ben.
In: Duke Mathematical Journal, Vol. 137, No. 1, 15.03.2007, p. 159-184.Research output: Contribution to journal › Article
TY - JOUR
T1 - Energy functionals and canonical Kähler metrics
AU - Song, Jian
AU - Weinkove, Ben
PY - 2007/3/15
Y1 - 2007/3/15
N2 - You [Y2] has conjectured that a Fano manifold admits a Kähler-Einstein metric if and only if it is stable in the sense of geometric invariant theory. There has been much progress on this conjecture by Tian [T1], [T2], Donaldson [Do1], [Do2], and others. The Mabuchi energy functional plays a central role in these ideas. We study the Ek functionals introduced by X. X. Chen and G. Tian [CT1] which generalize the Mabuchi energy. We show that if a Fano manifold admits a Kähler-Einstein metric, then the functional E1 is bounded from below and, modulo holomorphic vector fields, is proper. This answers affirmatively a question raised by Chen [C2]. In fact, we show that E1 is proper if and only if there exists a Kähler-Einstein metric, giving a new analytic criterion for the existence of this canonical metric, with possible implications for the study of stability. We also show that on a Fano Kähler-Einstein manifold, all of the functionals Ek are bounded below on the space of metrics with nonnegative Ricci curvature.
AB - You [Y2] has conjectured that a Fano manifold admits a Kähler-Einstein metric if and only if it is stable in the sense of geometric invariant theory. There has been much progress on this conjecture by Tian [T1], [T2], Donaldson [Do1], [Do2], and others. The Mabuchi energy functional plays a central role in these ideas. We study the Ek functionals introduced by X. X. Chen and G. Tian [CT1] which generalize the Mabuchi energy. We show that if a Fano manifold admits a Kähler-Einstein metric, then the functional E1 is bounded from below and, modulo holomorphic vector fields, is proper. This answers affirmatively a question raised by Chen [C2]. In fact, we show that E1 is proper if and only if there exists a Kähler-Einstein metric, giving a new analytic criterion for the existence of this canonical metric, with possible implications for the study of stability. We also show that on a Fano Kähler-Einstein manifold, all of the functionals Ek are bounded below on the space of metrics with nonnegative Ricci curvature.
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U2 - https://doi.org/10.1215/S0012-7094-07-13715-3
DO - https://doi.org/10.1215/S0012-7094-07-13715-3
M3 - Article
VL - 137
SP - 159
EP - 184
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
SN - 0012-7094
IS - 1
ER -