Collar Theorem Research Articles

Smoothed projections over weakly Lipschitz domains

We develop finite element exterior calculus over weakly Lipschitz domains. Specifically, we construct commuting projections from $L^p$ de Rham complexes over weakly Lipschitz domains onto finite element de Rham complexes. The projections satisfy uniform bounds for finite element spaces with bounded polynomial degree over shape-regular families of triangulations. Thus we extend the theory of finite element differential forms to polyhedral domains that are weakly Lipschitz but not strongly Lipschitz. As new mathematical tools, we use the collar theorem in the Lipschitz category, and we show that the degrees of freedom in finite element exterior calculus are flat chains in the sense of geometric measure theory.

Collars and partitions of hyperbolic cone-surfaces

For compact Riemann surfaces, the collar theorem and Bers’ partition theorem are major tools for working with simple closed geodesics. The main goal of this article is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic two-dimensional orbifolds are a particular case of such surfaces. We consider all cone angles to be strictly less than π to be able to consider partitions.

A Note on Collars of Simple Closed Geodesics

For compact hyperbolic Riemann surfaces, the collar theorem gives a lower bound on the distance between a simple closed geodesic and all other simple closed geodesics that do not intersect the initial geodesic. Here it is shown that there are two possible configurations, and in each configuration there is a natural collar width associated to a simple closed geodesic. If one extends the natural collar of a simple closed geodesic α by e >0, then the extended collar contains an infinity of simple closed geodesics that do not intersect α.

The riesz turndown collar theorem giving an asymptotic estimate of the powers of an operator under the ritt condition

For Banach space operatorsT satisfying the Tadmor-Ritt condition ‖(zI−T)−1‖≤C|z−1|−1, |z|>1, we show how to use the Riesz turndown collar theorem to estimate supn≥0‖Tn‖. A similar estimate is shown for lim supn ‖Tn‖ in terms of the Ritt constantM=lim supz→1‖(1−z)(zI−T)−1‖. We also obtain an estimate of the functional calculus for these operators proving, in particular, that ‖f(T)‖≤Cq‖f‖Mult, where ‖·‖Mult stands for the multiplier norm of the Cauchy-Stieltjes integrals over a Lusin type cone domain depending onC and a parameterq, 0<q<1.

Parabolics on the Boundary of the Deformation Space of a Kleinian Group

We present a condition on a loxodromic element L of a Kleinian group G which guarantees that L cannot be made parabolic on the boundary of the deformation space of G, namely, that the fixed points of L are separated by the limit set of a subgroup F of G which is a finitely generated quasifuchsian group of the first kind. The proof uses the collar theorem for short geodesics in hyperbolic 3-manifolds.

Horocycles on Riemann surfaces

By the Collar Theorem, every puncture on a hyperbolic Riemann surface with punctures has a horocyclic neighborhood of area 2 2 . Furthermore two such neighborhoods associated to different punctures are disjoint. This result can be improved if we omit the condition that horocyclic neighborhoods of different punctures must be disjoint. Using arguments of the second author we show, in this paper, that each puncture of a hyperbolic Riemann surface has a horocyclic neighborhood of area 4 4 .

An Open Collar Theorem for 4-manifolds

An Open Collar Theorem for 4-manifolds

An open collar theorem for 4-manifolds

Let M 4 {M^4} be an open 4 4 -manifold with boundary. Conditions are given under which M 4 {M^4} is homeomorphic to ∂ M × [ 0 , 1 ) \partial M \times [0,1) . Applications include a 4 4 -dimensional weak h h -cobordism theorem and a classification of weakly flat embeddings of 2 2 -spheres in S 4 {S^4} . Specific examples of ( n − 2 ) (n - 2) -spheres embedded in S n {S^n} (including n = 4 n = 4 ) are also discussed.

The collar theorem and examples

We generalize Randol's estimate of the length of cylinders in Riemann surfaces to arbitrary variable curvature and give examples with constant curvature to show that the bound is sharp.