arXiv research

Ancient curves span halfplanes via flow.

We consider minimal graphs u(x,y)>0 over unbounded domains D (with u vanishing on the boundary of D). Assuming D contains a sector properly containing a halfplane, we obtain estimates on growth and provide examples illustrating a range of growth.

In this paper we prove that any Riemannian surface, with no restriction of curvature at all, can be decomposed into blocks belonging just to some of these types: generalized Y-pieces, generalized funnels and halfplanes.

We develop the Lorentzian geometry of a crooked halfspace in 2+1-dimensional Minkowski space. We calculate the affine, conformal and isometric automorphism groups of a crooked halfspace, and discuss its stratification into orbit types, giving an explicit slice for the action of the automorphism group. The set of parall…

New Weyl's laws discovered for compact spaces with Ricci curvature bounds.

Every closed hyperbolic geodesic $γ$ on the triply--punctured sphere $M =\widehat{\mathbb C} - \{0,1,\infty\}$ has a self--intersection number $I(γ) \ge 1$ and a combinatorial length $L(γ) \ge 2$, the latter defined by the number of times $γ$ passes through the upper halfplane. In this paper we show that $δ(γ) = I(γ) -…

We investigate a problem posed by L. Hauswirth, F. Hélein, and F. Pacard, namely, to characterize all the domains in the plane that admit a "roof function", i.e., a positive harmonic function which solves simultaneously a Dirichlet problem with null boundary data, and a Neumann problem with constant boundary data. Unde…

Study reveals fundamental group properties of manifolds with specific curvature and growth.

In 1988, Karcher generalized the family of singly periodic Scherk minimal surfaces by constructing, for each natural $n\geq 2$, a $(2n-3)$-parameter family of singly periodic minimal surfaces with genus zero and $2n$ Scherk-type ends in the quotient, called {\it saddle towers}. They have been recently classified by Pér…

Paper disproves potential singularity models for 3D hypersurfaces in R^4.

Paper proves a relative version of coarse Alexander duality and applies it to Jordan cycles.

Four constructions of Seifert surfaces - Hopf plumbing, arborescent plumbing, basketry, and T-bandword handle decomposition - are described, and some interrelationships found, e.g.: arborescent Seifert surfaces are baskets; Hopf-plumbed baskets are precisely homogeneous T-bandword surfaces. A Seifert surface is Hopf-pl…