In this paper, we will establish a regularity theory for the Kähler–Ricci flow on Fano$n$-manifolds with Ricci curvature bounded in$L$^{$p$}-norm for some $${p > n}$$ . Using this regularity theory, we will also solve a long-standing conjecture for dimension 3. As an application, we give a new proof of the Yau–Tian–Donaldson conjecture for Fano 3-manifolds. The results have been announced in [45].

We give a complete classification of irreducible symmetric spaces for which there exist proper SL(2,R)-actions as isometries, using the criterion for proper actions by T. Kobayashi [Math. Ann. ’89] and combinatorial techniques of nilpotent orbits. In particular, we classify irreducible symmetric spaces that admit surface groups as discontinuous groups, combining this with Benoist’s theorem [Ann. Math. ’96].

We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming smallness of first derivatives of the defining map. An immediate application is a convergence theorem of the mean curvature flow of the graph of an area decreasing map between flat Riemann surfaces.

A variational formula for the Lutwak affine surface areas j of convex bodies in Rn is established when 1 ≤ j ≤ n − 1. By using introduced new ellipsoids associated with projection functions of convex bodies, we prove a sharp isoperimetric inequality for j , which opens up a new passage to attack the longstanding Lutwak conjecture in convex geometry.

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