We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in ${ℝ}^n$ with exponential non-linearity and target a convex body $P$ is solvable iff $0$ is the barycenter of $P.$ Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties $(X,\Delta )$ saying that $(X,\Delta )$ admits a (singular) Kähler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and...

By exploiting Perelman’s pseudolocality theorem, we prove a new compactness theorem for Ricci flows. By optimising the theory in the two-dimensional case, and invoking the theory of quasiconformal maps, we establish a new existence theorem which generates a Ricci flow starting at an arbitrary incomplete metric, with Gauss curvature bounded above, on an arbitrary surface. The criterion we assert for well-posedness is that the flow should be complete for all positive times; our discussion of uniqueness...

We investigate a coupled system of the Ricci flow on a closed manifold $M$ with the harmonic map flow of a map $\phi$ from $M$ to some closed target manifold $N$,$$\frac{\partial }{\partial t}g=-2\mathrm{Rc}+2\alpha \nabla \phi \otimes \nabla \phi ,\frac{\partial }{\partial t}\phi ={\tau }_g\phi ,$$where $\alpha$ is a (possibly time-dependent) positive coupling constant. Surprisingly, the coupled system may be less singular than the Ricci flow or the harmonic map flow alone. In particular, we can always rule out energy concentration of $\phi$ a-priori by choosing $\alpha$ large enough. Moreover, it suffices to bound the curvature of $(M,g(t\left)\right)$ to also obtain control of ...