**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