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We prove the longtime existence and convergence of the Calabi flow on toric Fano surfaces in a large family of Kahler classes where the class has positive extremal Hamiltonian potential and the initial Calabi energy is bounded by some constant. This is an extension of our previous work. We use the toric condition in a more essential way to rule out bubbles.
We study the generalized Kahler-Ricci flow with initial data of symplectic type, and show that this condition is preserved. In the case of a Fano background with toric symmetry, we establish global existence of the normalized flow. We derive an exten
We consider the formation of singularities along the Calabi flow with the assumption of the uniform Sobolev constant. In particular, on Kahler surface we show that any maximal bubble has to be a scalar flat ALE Kahler metric. In some certain classes
Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kahler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of
We recast the Calabi flow in DeGiorgis language of minimizing movements. We establish the long time existence of minimizing movements for K-energy with arbitrary initial condition. Furthermore we establish some a priori regularity of these solutions,
In this paper, we will study harmonic functions on the complete and incomplete spaces with nonnegative Ricci curvature which exhibit inhomogeneous collapsing behaviors at infinity. The main result states that any nonconstant harmonic function on such