Let $M = \sum_1 \times \sum_2 $ be the product of two compact Riemannian manifolds of dimension $n \ge 2$ and two, respectively. Let $\sum $ be the graph of a smooth map $f: \sum_1 \to \sum_2$, then $\sum$ is an n-dimensional submanifold of $M$. Let $\mathcal{G}$ be the Grassmannian bundle over $M$ whose fiber at each point is the set of all n-dimensional subspaces of the tangent space of $M$. The Gauss map $\gamma :\sum \to \mathcal{G}$ assigns to each point $x \in \sum $ the tangent space of \sum$ at x. This article considers the mean curvature flow of $\sum $ in $M$. When $\sum_1|$ and $\sum_2|$ are of the same non-negative curvature, we show a sub-bundle \mathcal{S}$ of the Grassmannian bundle is preserved along the flow, i.e. if the Gauss map of the initial submanifold $\sum $ lies in \mathcal{S}$, then the Gauss map of $\sum_t$ at any later time $t$ remains in $\mathcal{S}$. We also show that under this initial condition, the mean curvature flow remains a graph, exists for all time and converges to the graph of a constant map at infinity . As an application, we show that if $f$ is any map from $S^n$ to $S^2$ and if at each point, the restriction of $df$ to any two dimensional subspace is area decreasing, then f is homotopic to a constant map.

We show that the logarithmic derivatives of the convolution heat kernels on a uni-modular Lie group are exponentially integrable. This result is then used to prove an “integrated” Harnack inequality for these heat kernels. It is shown that this integrated Harnack inequality is equivalent to a version of Wang’s Harnack inequality. (A key feature of all of these inequalities is that they are dimension independent.) Finally, we show these inequalities imply quasi-invariance properties of heat kernel measures for two classes of infinite dimensional “Lie” groups.

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In this paper we explicitly construct Moishezon twistor spaces on nCP2 for arbitrary n ≥ 2 which admit a holomorphic Caction. When n = 2, they coincide with Y. Poon’s twistor spaces. When n = 3, they coincide with the ones studied by the author in [14]. When n ≥ 4, they are new twistor spaces, to the best of the author’s knowledge. By investigating the anticanonical system, we show that our twistor spaces are bimeromorphic to conic bundles over certain rational surfaces. The latter surfaces can be regarded as orbit spaces of the C-action on the twistor spaces. Namely they are minitwistor spaces. We explicitly determine their defining equations in CP4. It turns out that the structure of the minitwistor space is independent of n. Further, we explicitly construct a CP2-bundle over the resolution of this surface, and provide an explicit defining equation of the conic bundles. It shows that the number of irreducible components of the discriminant locus for the conic bundles increases as n does. Thus our twistor spaces have a lot of similarities with the famous LeBrun twistor spaces, where the minitwistor space CP1 ×CP1 in LeBrun’s case is replaced by our minitwistor spaces found in [15].

Let (Mn, g) be a complete non-compact K¨ahler manifold with non-negative and bounded holomorphic bisectional curvature. We prove that M is holomorphically covered by a pseudoconvex domain in Cn which is homeomorphic to R2n, provided (Mn, g) has uniform linear average quadratic curvature decay.