arXiv research

The paper defines and proves the existence of curves in Riemannian manifolds with prescribed angles to torse-forming vector fields.

Curve diffusion flow straightens curves with endpoints on intersecting lines.

The paper studies prescribed angle surfaces in Riemannian manifolds with torse-forming vector fields.

We consider planar networks of three curves that meet at two junctions with prescribed equal angles, minimizing a combination of the elastic energy and the length functional. We prove existence and regularity of minimizers, and we show some properties of the minimal configurations.

Study of curves and surfaces in Riemannian spaces making a constant angle with a parallel transported direction.

Mean curvature flow converges to a translating soliton with prescribed contact angle.

We show short-time existence for curves driven by curve diffusion flow with a prescribed contact angle $α\in (0, π)$: The evolving curve has free boundary points, which are supported on a line and it satisfies a no-flux condition. The initial data are suitable curves of class $W_2^γ$ with $γ\in (\tfrac{3}{2}, 2]$. For …

Spherical quadrilaterals classified based on geometric properties.

We study the prescribed mean curvature equation with a prescribed boundary contact angle condition in $M\times\R$ where $M^n$ is a Riemannian submanifold in $\R^{n+1}$. The main purpose is to establish a priori gradient estimates for solutions, from which the long time existence of the solution are derived.

Reproves results on spherical metrics using parabolic bundles.

Paper proves minimizing movements match smooth droplet flow in 3D.

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with …

Characterizes representations for complex projective structures with specific branch data.

Shows smoothness of varifolds with specific boundary angles.

Study proves existence of weak mean curvature flow with contact angle.

In this paper we study nonparametric mean curvature type flows in $M\times\mathbb{R}$ which are represented as graphs $(x,u(x,t))$ over a domain in a Riemannian manifold $M$ with prescribed contact angle. The speed of $u$ is the mean curvature speed minus an admissible function $ψ(x,u,Du)$. Long time existence and unif…

Investigates polar tangential angles of curves and their monotonicity.

In his paper "Shapes of Polyhedra and Triangulations of the Sphere", Thurston found that the set of shapes of convex polyhedra with prescribed cone-deficits has a complex hyperbolic structure. Inspired by his work, this paper studies the set of shapes of centrally symmetric octahedra with prescribed cone-deficits. We s…

Unified method visualizes curvature on curves and surfaces.

Proves existence and uniqueness of metrics with negative curvature and singularities on compact surfaces.

New approach to prescribing Gaussian curvature on spheres with conical singularities.

We show that for given four points on the sphere and prescribed angles at these points, which are not multiples of $2π$, the number of metrics of curvature 1 having conic singularities with these angles at these points is finite.

We introduce the class of perturbed right-angled Artin groups. These are constructed by gluing Bieri double groups into standard right-angled Artin groups. As a first application of this construction we obtain families of CAT(0) groups containing finitely presented subgroups which are not of type $\mathrm{FP}_3$, and h…

Optical interpretation of Euler's angle problem for caustics of light rays.

Smooth curves with specific curvature can be closely approximated.

We establish the short-time existence of the Ricci flow on surfaces with a finite number of conic points, all with cone angle between 0 and $2π$, where the cone angles remain fixed or change in some smooth prescribed way. For the angle-preserving flow we prove long-time existence and convergence. When the Troyanov angl…

The paper constructs surfaces with prescribed mean curvature in a specific space.

Sharp inequalities for curved surfaces and cones.

Study circle patterns and polyhedral surfaces in hyperbolic ends, proving manifold properties.

In this article we give a criterion for the existence of a metric of curvature $1$ on a $2$-sphere with $n$ conical singularities of prescribed angles $2π\vartheta_1,\dots,2π\vartheta_n$ and non-coaxial holonomy. Such a necessary and sufficient condition is expressed in terms of linear inequalities in $\vartheta_1,\dot…

We establish a connection between capillary floating in neutral equilibrium and the billiard ball problem. This allows us to reduce the question of floating in neutral equilibrium at any orientation with a prescribed contact angle for infinite homogeneous cylinders to a question about billiard caustics for their orthog…

A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.

Embedding right-angled Artin groups in mapping class groups of nonorientable surfaces.

Solves Jenkins-Serrin problem in 3-manifolds with Killing vector fields.

Classification of surfaces with prescribed mean curvature in Heisenberg space and SL2(R).

We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we …

The paper studies mean curvature flow with contact angles in high-dimensional cylinders.

Survey on metrics with conic singularities on Riemann surfaces.

Study on surfaces with conical singularities and geodesic boundaries, deriving existence results.

Paper explores folding patterns of curved creases preserving their geometric properties.

We prove that there exist solutions for a non-parametric capillary problem in a wide class of Riemannian manifolds endowed with a Killing vector field. In other terms, we prove the existence of Killing graphs with prescribed mean curvature and prescribed contact angle along its boundary. These results may be useful for…

Researchers solved a geometry paradox for creased tubes.

We consider an evolving plane curve with two endpoints that can move freely on the $x$-axis with generating constant contact angles. We discuss the asymptotic behavior of global-in-time solutions when the evolution of this plane curve is governed by area-preserving curvature flow equation. The main result shows that an…

The study sets limits on dihedral angles of large hyperbolic polyhedra.

Curves inscribe rectangles with positive area.

The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-spac…

In this paper we classify certain special ruled surfaces in $\R^3$ under the general theorem of characterization of constant angle surfaces. We study the tangent developable and conical surfaces from the point of view the constant angle property. Moreover, the natural extension to normal and binormal constant angle sur…

The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure …