We study the Emden-Fowler equation −∆u = |u|u on the hyperbolic space H . We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is p = (n + 2)/(n − 2) as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers [23, 4] con...

We show a Cr connecting lemma for area-preserving surface diffeomorphisms and for periodic Hamiltonian on surfaces. We prove that for a generic Cr , r = 1, 2, . . ., ∞, area-preserving diffeomorphism on a compact orientable surface, homotopic to identity, every hyperbolic periodic point has a transversal homoclinic point. We also show that for a Cr, r = 1, 2, . . ., ∞ generic time periodic Hami...

We define and compute hyperbolic coordinates and associated foliations which provide a new way to describe the geometry of the standard map. We also identify a uniformly hyperbolic region and a complementary “critical” region containing a smooth curve of tangencies between certain canonical “stable” foliations.

In the unit disk D hyperbolic geodesic rays emanating from the origin and hyperbolic disks centered at the origin exhibit simple geometric properties. The goal is to determine whether analogs of these geometric properties remain valid for hyperbolic geodesic rays and hyperbolic disks in a simply connected region Ω. According to whether the simply connected region Ω is a subset of the unit disk ...

Constructing smooth freeform surfaces of arbitrary topology with higher order continuity is one of the most fundamental problems in shape and solid modeling. Most real-world surfaces are with negative Euler characteristic number χ = 2 − 2g − b < 0, where g is genus, b is the number of boundaries. This paper articulates a novel method to construct C∞ smooth surfaces with negative Euler numbers b...

Constructing smooth freeform surfaces of arbitrary topology with higher order continuity is one of the most fundamental problems in shape and solid modeling. Most real-world surfaces are with negative Euler characteristic χ < 0. This paper articulates a novel method to construct C∞ smooth surfaces with negative Euler numbers based on hyperbolic geometry and discrete curvature flow. According to...