Description

This mini course aims to give a brief introduction to K\"ahler-Ricci (KR) flow and KR solitons on Fano manifolds. We will present a short proof of the convergence of KR flow on manifolds admitting a K\"ahler-Einstein (KE) metric or KR soliton.

Lecture 1: we will discuss basics on complex Monge-Amp\`ere equation and the related a priori estimates \cite{Y, K, GPT}. The relation between the existence of KE metrics and lower bounds of certain eigenvalues will be covered \cite{GPS2, PSSW}.

Lecture 2: we will start with Perelman's uniform estimates along Kahler-Ricci flow and present a short proof (c.f. \cite{GPS1}) of the smooth convergence of K\"ahler-Ricci flow on Fano manifolds admitting a KE metric or KR soliton, with the help the Moser-Trudinger inequality \cite{PSSW1,DR}. This result is first proved by Perelman and Tian-Zhu.

Lecture 3: motivated by the compactness of KE metrics, we discuss the compactness of Kahler-Ricci solitons on Fano manifolds \cite{GPSS}. In particular, the uniform bound on the Futaki invariants on Fano manifolds will be derived and this plays an important role in the proof.

Prerequisite

No specific advanced knowledge is needed, but it would be helpful to be familiar with the basics of complex geometry and linear partial differential equations.

References

[DR] Darvas T. and Y. Rubinstein,  Tian’s properness conjectures and Finsler geometry of the space of K\"ahler metrics. J. Amer. Math. Soc. 30 (2017), 347 – 387.

[GPSS] Guo, B., Phong, D. H., Song, J. and Sturm, J., Compactness of K\"ahler-Ricci solitons on Fano manifolds, arXiv:1805.03084v1

[GPS1] Guo, B., Phong, D. H. and Sturm, J. On the K\"ahler-Ricci flow on Fano manifolds, arXiv:2001.06329v1

[GPS2] Guo, B., Phong, D. H. and Sturm, J. K\"ahler-Einstein Metrics and Eigenvalue Gaps, arXiv:2001.05794v1

[GPT] Guo, B., Phong, D. H. and Tong, F., On $L^\infty$ estimates for complex Monge-Amp\`ere equations, arXiv:2106.02224

[K] Kolodziej, S. H\"older continuity of solutions to the complex Monge-Amp\`ere equation with the right hand side in $L^p$. The case of compact Kahler manifolds. Math. Ann. 342, 379 - 386 (2008)

[PSSW] Phong, D. H., J. Song, J. Sturm, and B. Weinkove, The K\"ahler-Ricci flow and the $\bar \partial$ operator on vector fields. J. Differential Geom. 81 (2009), no. 3, 631 - 647.

[PSSW1] Phong, D. H., J. Song, J. Sturm, and B. Weinkove, The Moser-Trudinger inequality on K\"ahler-Einstein manifolds, Amer. J. of Math. 130 no 4 (2008) 1067-1085.

[Y] Yau, S.T., On the Ricci curvature of a compact K\"ahler manifold and the complex Monge-Amp\`ere equation. I. Comm. Pure Appl. Math. 31 (1978), no. 3, 339 - 411

Notes:

 Lecture 1- KE and eigenvalue gap.pdf  Lecture 2- KE and eigenvalue gap, Kahler-Ricci flow.pdf  Lecture 3- compactness of Kahler-Ricci solitons.pdf