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.

In this article we show how the classical theory of Reeb and others extends to the case of codimension one singular foliations on closed oriented manifolds, provided that at the singular set sing(F), the foliation is locally defined by Bott-Morse functions which are transversely centers. We prove, in this setting, the equivalent of the local and the complete stability theorems of Reeb. We show that if F has a compact leaf with finite fundamental group, or if a component of sing(F) has codimension ≥ 3 and finite fundamental group, then all leaves of F are compact and diffeomorphic, sing(F) consists of two connected components, and there is a Bott-Morse function f : M → [0, 1] such that f : M \ sing(F) → (0, 1) is a fiber bundle defining F and sing(F) = f.1({0, 1}). This yields a topological description of the type of leaves that appear in these foliations, and also the type of manifolds admitting such foliations. These results unify and generalize well known results for cohomogeneity one isometric actions, and a theorem of Reeb for foliations with Morse singularities of center type.

In this note we introduce a natural Finsler structure on convex surfaces, referred to as the quotient Finsler structure, which is dual in a sense to the inclusion of a convex surface in a normed space as a submanifold. It has an associated quotient girth, which is similar to the notion of girth defined by Sch¨affer. We prove the analogs of Sch¨affer’s dual girth conjecture (proved by ´ Alvarez-Paiva) and the Holmes–Thompson dual volumes theorem in the quotient setting. We then show that the quotient Finsler structure admits a natural extension to higher Grassmannians, and prove the corresponding theorems in the general case. We follow ´ Alvarez-Paiva’s approach to the problem, namely, we study the symplectic geometry of the associated co-ball bundles. For the higher Grassmannians, the theory of Hamiltonian actions is applied.

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The Cauchy problem for the homogeneous real/complexMongeAmp`ere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of K¨ahler metrics equipped with the Mabuchi metric. This Cauchy problem is believed to be ill-posed and a basic problem is to characterize initial data of (weak) solutions which exist up to time T . In this article, we use a quantization method to construct a subsolution of the HCMA on a general projective variety, and we conjecture that it solves the equation for as long as the unique solution exists. The subsolution, called the “quantum analytic continuation potential,” is obtained by (i) Toeplitz quantizing the Hamiltonian flow determined by the Cauchy data, (ii) analytically continuing the quantization, and (iii) taking a certain logarithmic classical limit. We then prove that in the case of torus invariant metrics (where the HCMA reduces to the HRMA) the quantum analytic continuation potential coincides with the well-known Legendre transform potential, and hence solves the equation as long as it is smooth. In the sequel [29], it is proved that the Legendre transform potential ceases to solve the HCMA once it ceases to be smooth. The results here and in the sequels in particular characterize the initial data of smooth geodesic rays.