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 study minimizers of the functionalwhere $B_{1}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .1ex\hbox {$\scriptscriptstyle +$}} {\scriptscriptstyle +}}}=\{x\in B_{1}: x_{1}>0\}$ ,$u$=0 on {$x$∈$B$_{1}:$x$_{1}=0}, $\lambda^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle \pm $}} {\raise .17ex\hbox {$\scriptstyle \pm $}} {\raise .1ex\hbox {$\scriptscriptstyle \pm $}} {\scriptscriptstyle \pm }}}$ are two positive constants and 0<$p$<1. In two dimensions, we prove that the free boundary is a uniform$C$^{1}graph up to the flat part of the fixed boundary and also that two-phase points cannot occur on this part of the fixed boundary. Here, the free boundary refers to the union of the boundaries of the sets {$x$:±$u$($x$)>0}.

We show that expanding Kahler-Ricci solitons which have positive holomorphic bisectional curvature and are C^2-asymptotic to a conical Kahler manifold at infinity must be the U(n)-rotationally symmetric expand solitons constructed by Cao.

We study the Plateau Problem of finding an area minimizing disk bounding a given Jordan curve in Alexandrov spaces with curvature ≥ κ. These are complete metric spaces with a lower curvature bound given in terms of triangle comparison. Imposing an additional condition that is satisfied by all Alexandrov spaces according to a conjecture of Perel’man, we develop a harmonic map theory from two dimensional domains into these spaces. In particular, we show that the solution to the Dirichlet problem from a disk is H¨older continuous in the interior and continuous up to the boundary. Using this theory, we solve the Plateau Problem in this setting generalizing classical results in Euclidean space (due to J. Douglas and T. Rado) and in Riemannian manifolds (due to C.B. Morrey).

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