arXiv research

Hot spots conjecture proven for small eigenvalue domains.

Study visibility properties of Kobayashi distance on unbounded domains.

Sharp bounds for spanning tree entropy in planar lattices.

New inequalities for planar convex domains' Laplacian eigenvalues.

One-parameter hyperbolic planar motion was first studied by S. Y$\ddot{\texttt{u}}$ce and N. Kuruo$\tilde{\texttt{g}}$lu. Moreover, they analyzed the relationships between the absolute, relative and sliding velocities of one-parameter hyperbolic planar motion as well as the related pole curves, \cite{Yuc}. One-paramete…

The aim of this paper is to clarify the relationship between Gromov-hyperbolicity and amenability for planar maps.

We define and study a Möbius invariant energy associated to planar domains, as well its generalization to space curves. This generalization is a Möbius version of Banchoff-Pohl's notion of area enclosed by a space curve. A relation with Gauss-Bonnet theorems for complete surfaces in hyperbolic space is also described.

Study on hyperbolic knotoids, proving their volumes add and providing tables.

In this paper we give two examples of sequences of embedded minimal planar domains in $\mathbb{R}^3$ which converge to singular laminations of $\mathbb{R}^3$. In contrast with the situation for embedded minimal disks, these examples do not arise from complete embedded minimal planar domains and highlight some of the su…

Study on planar graphs in Poincare model of hyperbolic geometry.

Paper extends tree bijection for hyperbolic surfaces without requiring cusps.

Estimates the index of the Laplace operator on planar domains with Robin boundary condition.

We initiate the study of the higher-order Escobar constants $I_k(M)$, $k\geq 3$, on bounded planar domains $M$. The Escobar constants $I_k$ of the unit disk and a family of polygons are provided.

In this paper, by the method of moving planes, we establish the monotonicity and symmetry properties of convex solutions for Monge-Ampere systems on bounded smooth planar domains.

We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the cor…

Study of pursuit-evasion game on sphere and its relation to planar Apollonius circle.

We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in finite time; then it shrinks homothetically to a point. Our theorem is the analog of the result of Grayson for curvature flow of clo…

In \cite{Mul} one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane $\mathbb{E}^2$ were obtained. Moreover, the relations between the Complex velocities one-parameter motion in the Complex plane were provided by \cite…

We prove that any non-simply connected planar domain can be properly and minimally embedded in H^2 x R. The examples that we produce are vertical bi-graphs, and they are obtained from the conjugate surface of a Jenkins-Serrin graph.

We give a necessary and sufficient condition for a hyperbolic Coxeter group with planar nerve to have Sierpiński curve as its Gromov boundary.

Isothermic tori with one planar curvature line found and characterized.

The Blaschke rolling disk theorem is extended to non-convex domains.

The study finds lower bounds for the first eigenvalue of the Laplacian in planar domains with magnetic fields.

In this note, we exhibit infinite families of tight non-fillable contact manifolds supported by planar open books with vanishing Heegaard Floer contact invariants. Moreover, we also exhibit an infinite such family where the supported manifold is hyperbolic.

The study finds counterexamples to a conjecture about incompressible planar surfaces in hyperbolic link exteriors.

The paper discusses fractional Sobolev immersions of flat domains into 3D space.

Proves convergence groups on a 2-sphere are Kleinian groups.

Upper bounds for magnetic Laplacian eigenvalues on planar domains.

Infinite-genus surfaces have many isospectral hyperbolic structures.

Reconstructing a planar domain from its Dirichlet-to-Neumann data

In 1997, Collin proved that any properly embedded minimal surface in $\mathbb{R}^3$ with finite topology and more than one end has finite total Gaussian curvature. Hence, by an earlier result of Lopez and Ros, catenoids are the only non-planar, non-simply connected, properly embedded, minimal planar domains in $\mathbb…

The paper classifies vertices in planar polygons formed by convex domains.

Study of group boundaries and subgroup properties.

The Fridman function is bounded by the injectivity radius for certain hyperbolic manifolds.

In this expository note, we illustrate phenomena and conjectures about boundaries of hyperbolic groups by considering the special cases of certain amalgams of hyperbolic groups. While doing so, we describe fundamental results on hyperbolic groups and their boundaries by Bowditch and Haissinsky, along with special treat…

We extend our discrete uniformization theorems for planar, $m$-connected, Jordan domains [Journal für die reine und angewandte Mathematik 670 (2012), 65--92] to closed surfaces of non-positive genus.

An ideal triangulation $\mathcal{T}$ of a hyperbolic 3-manifold $M$ with one cusp is non-peripheral if no edge of $\mathcal{T}$ is homotopic to a curve in the boundary torus of $M$. For such a triangulation, the gluing and completeness equations can be solved to recover the hyperbolic structure of $M$. A planar project…

Defines W-volume for planar domains with circular boundaries, relating to Laplacian determinant and Schottky uniformization.

New bounds on inscribed triangles in arbitrary planar domains.

We deduce from a rooted tree in the disk a slalom divide and a slalom knot. A slalom knot is either the local link of a simple plane curve singularity of type A_2n, E_6, E_8 or a fibered hyperbolic knot with very special monodromy.

We study an explicit construction of planar open books with four binding components on any three-manifold which is given by integral surgery on three component pure braid closures. This construction is general, indeed any planar open book with four binding components is given this way. Using this construction and resul…

New proof shows not all Salem numbers are growth rates of Coxeter groups.

Study stable capillary hypersurfaces with planar boundaries in half-spaces and domains.

These notes outline recent developments in classical minimal surface theory that are essential in classifying the properly embedded minimal planar domains M in R^3 with infinite topology (equivalently, with an infinite number of ends). This final classification result by Meeks, Perez, and Ros states that such an M must…

The paper characterizes sets with infinite hyperbolic convex hull volume.

Random hyperbolic surfaces with punctures converge to the Brownian sphere.

Fix a finite set of points in Euclidean $n$-space $\euc^n$, thought of as a point-cloud sampling of a certain domain $D\subset\euc^n$. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of $D$. …

We provide examples of towers of covers of cusped hyperbolic 3-manifolds whose exponential homological torsion growth is explicitly computed in terms of volume growth. These examples arise from abelian covers of alternating links in the thickened torus. A corollary is that the spanning tree entropy for each regular pla…