We consider reparametrizations of Heisenberg nilflows.We show that if a Heisenberg nilflow is uniquely ergodic, all non-trivial timechanges within a dense subspace of smooth time-changes are mixing. Equivalently, in the language of special flows, we consider special flows over linear skew-shift extensions of irrational rotations of the circle. Without assuming any Diophantine condition on the frequency, we define a dense class of smooth roof functions for which the corresponding special flows are mixing whenever the roof function is not a coboundary. Mixing is produced by a mechanism known as stretching of ergodic sums. The complement of the set of mixing time-changes (or, equivalently, of mixing roof functions) has countable codimension and can be explicitly described in terms of the invariant distributions for the nilflow (or, equivalently,for the skew-shift), producing concrete examples of mixing time-changes.

In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature dimension inequality C D E(n, K), which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, heat kernel bounds and Cheng's eigenvalue estimate. These are the first kind of results in this direction for unbounded Laplacian on graphs.

We introduce a new geometric flow, called the chord shortening flow, which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we give a simplified proof of a classical theorem of Lusternik and Schnirelmann (and a generalization by Riede and Hayashi) on the existence of multiple orthogonal geodesic chords. For a compact convex planar domain, we show that any convex chord not orthogonal to the boundary would shrink to a point in finite time under the flow.

We characterize H¨older continuously differentiable m dimensional submanifolds of Euclidean space among m rectifiable sets S in terms of growth conditions on the m density ratios of the Hausdorff measure Hm S.

This article considers the quasi-local conserved quantities with respect to a reference spacetime with a cosmological constant. We follow the approach developed by the authors in [25,26,7] and define the quasi-local energy as differences of surface Hamiltonians. The ground state for the gravitational energy is taken to be a reference configuration in the de Sitter (dS) or Anti-de Sitter (AdS) spacetime. This defines the quasi-local energy with respect to the reference spacetime and generalizes our previous definition with respect to the Minkowski spacetime. Through an optimal isometric embedding into the reference spacetime, the Killing fields of the reference spacetime are transplanted back to the surface in the physical spacetime to complete the definitions of quasi-local conserved quantities. We also compute how the corresponding total conserved quantities evolve under the Einstein equation with a cosmological constant.