arXiv research

Compact Special Weingarten surfaces with planar convex boundaries are disks.

We study the Dirichlet problem for a graph $Σ$ in $\mathbb{R}^{n+1}$ with normalized constant mean curvature $H>0$ and planar boundary $Γ=\partial Ω$. Our main result is that the optimal solvability condition, namely that the normalized mean curvature $h$ of $Γ$ satisfies $h\geq H$, also suffices when $Ω$ is strictly c…

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

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

We prove that maximal annuli in $\mathbb{L}^{3}$ bounded by circles, straight lines or cone points in a pair of parallel spacelike planes are part of either a Lorentzian catenoid or a Lorentzian Riemann's example. We show that under the same boundary condition, the same conclusion holds even when the maximal annuli hav…

Study on curve shortening flow with boundary conditions, proving convergence or contraction.

We prove that the curvature flow of an embedded planar network of three curves connected through a triple junction, with fixed endpoints on the boundary of a given strictly convex domain, exists smooth until the lengths of the three curves stay far from zero. If this is the case for all times, then the evolution exists…

New inequalities for planar convex domains' Laplacian eigenvalues.

Study on regularity of optimal transport maps on convex domains with quadratic cost.

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

We consider compact connected minimal surfaces, with a pair of boundary curves (not necessarily convex) in distinct planes, that have least-area amongst all orientable surfaces with the same boundary. When the planes containing these two boundary curves are either parallel or sufficiently close to parallel, and when th…

We investigate the planarity of the boundaries of right-angled Coxeter groups. We show that non-planarity of the defining graph does not necessarily imply non-planarity of every boundary of the associated right-angled Coxeter group, although it does in many cases. Our techniques yield a characterization of the triangle…

Study inverse boundary value problem for Monge-Ampère equation on convex domains.

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

Study constructs disks with curved boundaries in a 3D ball.

We prove that the only compact convex ancient solutions of the planar affine normal flow are contracting ellipses.

Proves planarity and convexity for ancient solutions of mean curvature flow.

Study the boundaries of ε-neighborhoods of planar sets, showing their structure and curvature.

The paper proves the exact number of singular points in the intersection of convex shapes.

Given a planar compact convex billiard table $T$, we give an algorithm to find the shortest generalised closed billiard orbits on $T$. (Generalised billiard orbits are usual billiard orbits if $T$ has smooth boundary.) This algorithm is finite if $T$ is a polygon and provides an approximation scheme in general. As an i…

Hot spots conjecture proven for small eigenvalue domains.

Proves existence of non-planar minimal disks in ellipsoids.

We show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid.

A classical combinatorial fact is that the simplicial complex consisting of disjointly embedded chords in a convex planar polygon is a sphere. For any surface F with non-empty boundary, there is an analogous complex Arc(F) consisting of suitable equivalence classes of arcs in F connecting its boundary components. The m…

The paper finds new inequalities for convex polygons.

A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass--Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian…

Study of minimal surfaces in a specific symmetric space with polynomial growth.

We determine the condition on a given lens space having a realization as a closure of homology cobordism over a planar surface with a given number of boundary components. As a corollary, we see that every lens space is represented as a closure of homology cobordism over a planar surface with three boundary components. …

The paper studies minimal surface flow and translating solitons, proving global solutions and convergence.

We introduce a new geometric flow called the chord shortening flow which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we give a simplified proof of a classical theorem of Lusternik and Schnirelmann (and…

The paper studies area-preserving and length-preserving inverse curvature flow for planar curves with singularities.

This article introduces planar ribbons, Vergili ribbon complexes and ribbon nerves in Alexandroff-Hopf-Whitehead CW (Closure finite Weak) topological spaces. A {\em planar ribbon} (briefly, {ribbon}) in a CW space is the closure of a pair of nesting, non-concentric filled cycles that includes the boundary but does not …

Framework for isometric immersions of planar regions from framed curves.

We develop two types of integral formulas for the perimeter of a convex body K in planar geometries. We derive Cauchy-type formulas for perimeter in planar Hilbert geometries. Specializing to H^2 we get a formula that appears to be new. We show that it implies the standard Cauchy-Santalo formula involving a central ang…

Proves Birkhoff-Poritsky conjecture for centrally-symmetric billiards.

Using open book foliations we show that an overtwisted disc in a planar open book can be put in a topologically nice position. As a corollary, we prove that a planar open book whose fractional Dehn twist coefficients grater than one for all the boundary components supports a tight contact structure.

New heat trace coefficients reveal curvature effects in polygonal domains.

Paper introduces length measures for curves and convex shapes, proving isoperimetric and distance properties.

Study of group boundaries and subgroup properties.

Study on planar graphs in Poincare model of hyperbolic geometry.

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 define a computable topological invariant $μ(γ)$ for generic closed planar regular curves $γ$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves witho…

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

We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that gr…

In this paper we study singular points of the Wigner caustic and affine $λ$--equidistants of planar curves based on shapes of these curves. We generalize the Blaschke-Süss theorem on the existence of antipodal pairs of a convex curve.

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.

Study three discrete envelope types of polygon bisection lines.

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