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.

In this paper, employing a new inequality, we show that under certain curvature pinching condition, the strictly convex closed smooth self-similar solution of $\sigma_k^{\alpha}$-flow must be a round sphere. We also obtain a similar result for the solutions of $F=-\langle X, e_{n+1}\rangle \, (*)$ with a non-homogeneous function $F$. At last, we prove that if $F$ can be compared with $\frac{(n-k+1)\sigma_{k-1}}{k\sigma_{k}}$, then a closed strictly $k$-convex solution of $(*)$ must be a round sphere.

We study minimal graphic functions on complete Riemannian manifolds $\Sigma$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with such growth via splitting for tangent cones of $\Sigma$ at infinity. When, in contrast, we do not impose any growth restrictions for minimal graphic functions, we also obtain a Liouville type theorem under a certain non-radial Ricci curvature decay condition on $\Sigma$. In particular, the borderline for the Ricci curvature decay is sharp by our example in the last section.

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A well-known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a zero-order pseudodifferential projection on a closed manifold vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in the odd dimensional case an analogous statement holds for the algebra of projective pseudodifferential operators, i.e., the noncommutative residue of a projective pseudodifferential projection vanishes. Our strategy of proof also simplifies the argument in the classical setting. We show that the noncommutative residue factors to a map from the twisted K-theory of the co-sphere bundle and then use arguments from twisted K-theory to reduce this question to a known model case. It can then be verified by local computations that this map vanishes in odd dimensions.