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Ends 7 Oct 2026 13:00
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KӒHLER-EINSTEIN METRICS WITH LARGE SYMMETRY - LECTURE 4
ICTP
Lagrange Lecture Hall
Abstract:
In 2004, Wang and I found a priori C°-estimates for certain real Monge-Ampère equations to study Kähler-Einstein metrics on toric manifolds. This method was later generalized by many people to study Kähler-Einstein metrics or Kähler-Ricci solitons on G-manifolds, horosymmetric manifolds, spherical varieties etc. Recently, Tian and I found a local version of the Wang-Zhu’s estimate to study the degeneration of Kähler-Ricci flow on such manifolds. This method can be also used to construct complete Kähler-Ricci solitons on certain non-compact toric manifolds. In this series of lectures, we will give four lectures related to the above works as follows:
Wed, 16 Sept: Lecture 1: Kähler-Einstein metrics on toric manifolds
Wed, 23 Sept: Lecture 2: Complete Kähler-Ricci solitons on non-compact toric manifolds
Wed, 30 Sept: Lecture 3: Some fundamental results and volume monotonicity on Kähler-Ricci flow on Fano manifolds
Wed, 7 Oct: Lecture 4: Degeneration of Kähler-Ricci flow on Fano G-manifolds
In 2004, Wang and I found a priori C°-estimates for certain real Monge-Ampère equations to study Kähler-Einstein metrics on toric manifolds. This method was later generalized by many people to study Kähler-Einstein metrics or Kähler-Ricci solitons on G-manifolds, horosymmetric manifolds, spherical varieties etc. Recently, Tian and I found a local version of the Wang-Zhu’s estimate to study the degeneration of Kähler-Ricci flow on such manifolds. This method can be also used to construct complete Kähler-Ricci solitons on certain non-compact toric manifolds. In this series of lectures, we will give four lectures related to the above works as follows:
Wed, 16 Sept: Lecture 1: Kähler-Einstein metrics on toric manifolds
Wed, 23 Sept: Lecture 2: Complete Kähler-Ricci solitons on non-compact toric manifolds
Wed, 30 Sept: Lecture 3: Some fundamental results and volume monotonicity on Kähler-Ricci flow on Fano manifolds
Wed, 7 Oct: Lecture 4: Degeneration of Kähler-Ricci flow on Fano G-manifolds
References:
[1] X. Wang and X.H. Zhu, Kähler-Ricci solitons on toric manifolds with positive first Chern class, Adv. Math., 188 (2004), 87-103.
[2] T. Delcroix, Kähler-Einstein metrics on group compactifications, Geom. Funct. Anal., 27 (2017), 78-129.
[3] N. Sesum and G. Tian, Bounding scalar curvature and diameter along the Kähler-Ricci flow (after Perelman), J. Inst. Math, Jussiu, 7 (2008), 575-587.
[4] J. Ye and X.H. Zhu, Weighted volume comparison and monotonicity for Lp-bound of Bakry-Emery Ricci curvature, arXiv:2604.17367.
[5] G. Tian and X.H. Zhu, Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds, J. Eur. Math. Soc., doi.org/10.417/JEMS/153, arXiv:2209.05029.