We introduce the notion of symplectic flatness for connections and fiber bundles over symplectic manifolds. Given an $A_\infty$-algebra, we present a flatness condition that enables the twisting of the differential complex associated with the $A_\infty$-algebra. The symplectic flatness condition arises from twisting the $A_\infty$-algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang-Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.

Let $M^n$ be a complete noncompact K\"ahler manifold with nonnegative bisectional curvature and maximal volume growth, we prove that $M$ is biholomorphic to $\mathbb{C}^n$. This confirms the uniformization conjecture of Yau under the assumption $M$ has maximal volume growth.

We give a complete classification of locally strongly convex affine hypersurfaces of Rn+1 with parallel cubic form with respect to the Levi-Civita connection of the affine Berwald-Blaschke metric. It turns out that all such affine hypersurfaces are quadrics or can be obtained by applying repeatedly the Calabi product construction of hyperbolic affine hyperspheres, using as building blocks either the hyperboloid, or the standard immersion of one of the symmetric spaces SL(m,R)/SO(m), SL(m,C)/SU(m), SU 2m /Sp(m), or E6(−26)/F4.

Existence and uniqueness of the solution to the Lp Minkowski problem for the electrostatic p-capacity are proved when p > 1 and 1 < p < n. These results are nonlinear extensions of the very recent solution to the Lp Minkowski problem for p-capacity when p = 1 and 1 < p < n by Colesanti et al. and Akman et al., and the classical solution to the Minkowski problem for electrostatic capacity when p = 1 and p = 2 by Jerison.

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