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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 on toric Fano surface, the Sobolev constant is a priori bounded along the Calabi flow with small Calabi energy. Also we can show in certain case no maximal bubble can form along the flow, it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently. To illustrate our results more clearly, we focus on an example on CP^2 blown up three points at generic position. Our result also implies existence of constant scalar curvature metrics on CP^2 blown up three points at generic position in the Kahler classes where the exceptional divisors have the same area.
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
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 formulate a Calabi-Yau type conjecture in generalized Kahler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for generalized Kahler structures generalizing the notion of Kahl
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
The scalar curvature equation for rotation invariant Kahler metrics on $mathbb{C}^n backslash {0}$ is reduced to a system of ODEs of order 2. By solving the ODEs, we obtain complete lists of rotation invariant zero or positive csck on $mathbb{C}^n ba