Suppose L is a lamination of a Riemannian manifold by hypersurfaces with the same constant mean curvature H. We prove that every limit leaf of L is stable for the Jacobi operator. A simple but important consequence of this result is that the set of stable leaves of L has the structure of a lamination.

In the first paper of this series, “Electrodynamics and the Gauss Linking Integral on the 3-sphere and in Hyperbolic 3-space,” we developed a steady-state version of classical electrodynamics in these two spaces, including explicit formulas for the vector-valued Green’s operator, explicit formulas of Biot-Savart type for the magnetic field, and a corresponding Amp`ere’s Law contained in Maxwell’s equations, and then used these to obtain explicit inte- gral formulas for the linking number of two disjoint closed curves. In this second paper, we obtain integral formulas for twisting, writhing, and helicity, and prove that link = twist + writhe on the 3-sphere and in hyperbolic 3-space. We then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of these two spaces. An announcement of these results, and a hint of their proofs, can be found in the Math ArXiv, math.GT/0406276, while an expanded version of the first paper, with full proofs, can be found at math.GT/0510388.

In this article, we study the small sphere limit of the Wang-Yau quasi-local energy defined in [18,19]. Given a point p in a spacetime N , we consider a canonical family of surfaces approaching p along its future null cone and evaluate the limit of the Wang-Yau quasi-local energy. The evaluation relies on solving an "optimal embedding equation" whose solutions represent critical points of the quasi-local energy. For a spacetime with matter fields, the scenario is similar to that of the large sphere limit found in [7]. Namely, there is a natural solution which is a local minimum, and the limit of its quasi-local energy recovers the stress-energy tensor at p . For a vacuum spacetime, the quasi-local energy vanishes to higher order and the solution of the optimal embedding equation is more complicated. Nevertheless, we are able to show that there exists a solution which is a local minimum and that the limit of its quasi-local energy is related to the Bel-Robinson tensor. Together with earlier work [7], this completes the consistency verification of the Wang-Yau quasi-local energy with all classical limits.

It is classically known that complete flat (that is, zero Gaussian curvature) surfaces in Euclidean 3-space R3 are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface f admits singularities but its Gauss map  is globally defined on the surface and can be smoothly extended across the singular set, f is called a frontal. In addition, if the pair (f, ) defines an immersion into R3×S2, f is called a front. A front f is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the caustic (i.e. focal surface) of a flat front f are also flat fronts. In this paper, we generalize the classical notion of completeness to flat fronts, and give a representation formula for a flat front which has a non-empty compact singular set and whose ends are all immersed and complete. As an application, we show that such a flat front has properly embedded ends if and only if its Gauss map image is a convex curve. Moreover, we show the existence of at least four singular points other than cuspidal edges on such a flat front with embedded ends, which is a variant of the classical four vertex theorem for convex plane curves.

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.