A foliation F on a Riemannian manifold M is hyperpolar if it admits a flat section, that is, a connected closed flat submanifold of M that intersects each leaf of F orthogonally. In this article we classify the hyperpolar homogeneous foliations on every Riemannian symmetric space M of noncompact type. These foliations are constructed as follows. Let  be an orthogonal subset of a set of simple roots associated with the symmetric space M. Then  determines a horospherical decomposition M = Fs ×ErankM−||×N, where Fs  is the Riemannian product of || symmetric spaces of rank one. Every hyperpolar homogeneous foliation on M is isometrically congruent to the product of the following objects: a particular homogeneous codimension one foliation on each symmetric space of rank one in Fs , a foliation by parallel affine subspaces on the Euclidean space ErankM−||, and the horocycle subgroup N of the parabolic subgroup of the isometry group of M determined by .

In this work, we give a perturbation theorem for strong polynomial solutions to the zero surface tension Hele-Shaw equation driven by injection or suction, the so called Polubarinova–Galin equation. This theorem enables us to explore properties of solutions with initial functions close to polynomials. Applications of this theorem are given in the suction and injection cases. In the former case, we show that if the initial domain is close to a disk, most of the fluid will be sucked before the strong solution blows up. In the latter case, we obtain precise large-time rescaling behaviors for large data to Hele-Shaw flows in terms of invariant Richardson complex moments. This rescaling behavior result generalizes a recent result regarding large-time rescaling behavior for small data in terms of moments. As a byproduct of a theorem in this paper, a short proof of existence and uniqueness of strong solutions to the Polubarinova–Galin equation is given.

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Let $ f : \sum_1 \to \sum_2$ be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in $\sum_1 \times \sum_2$2 by the mean curvature flow. Under suitable conditions on the curvature of $\sum_1$ and $\sum_2$ and the differential of the initial map, we show that the flow exists smoothly for all time. At each instant $t$, the flow remains the graph of a map $f_t$ and $f_t$ converges to a constant map as $t$ approaches infinity. This also provides a regularity estimate for Lipschtz initial data.