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A horospherical torus about a cusp of a hyperbolic manifold inherits a Euclidean similarity structure, called a cusp shape. We bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation preserving the cusp. The bounds are in terms of the change in structure in a neighborhood of the singular locus alone. We then apply this result to provide information on the cusp shape of many hyperbolic knots, given only a diagram of the knot. Specifically, we show there is a universal constant C such that if a knot admits a prime, twist reduced diagram with at least C crossings per twist region, then the length of the second shortest curve on the cusp torus is bounded. The bound is linear in the number of twist regions of the diagram.

Let (M, g) be a complete 3-dimensional asymptotically flat manifold with everywhere positive scalar curvature. We prove that, given a compact subset K  M, all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid K. This complements the results of G. Huisken and S.-T. Yau [17] and of J. Qing and G. Tian [26] on the uniqueness of large volume preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to Schwarzschild with mass m > 0. The analysis in [17] and [26] takes place in the asymptotic regime of M. Here we adapt ideas from the minimal surface proof of the positive mass theorem [32] by R. Schoen and S.-T. Yau and develop geometric properties of volume preserving stable constant mean curvature surfaces to handle surfaces that run through the part of M that is far from Euclidean.

In this note we show that one open 4-dimensional ellipsoid embeds symplectically into another if and only if the ECH capacities of the first are no larger than those of the second. This proves a conjecture due to Hofer. The argument uses the equivalence of the ellipsoidal embedding problem with a ball embedding problem that was recently established by McDuff. Its method is inspired by Hutchings’ recent results on embedded contact homology (ECH) capacities but does not use them.

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus, the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, we consider weighted Delaunay triangulations in piecewise flat surfaces, allowing cone singularities with prescribed cone angles in the vertices. The material presented here extends work by Rivin on Delaunay triangulations and ideal polyhedra.