arXiv research

New inequalities for planar convex domains' Laplacian eigenvalues.

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…

Hot spots conjecture proven for small eigenvalue domains.

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.

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…

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.

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.

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

Upper bounds for magnetic Laplacian eigenvalues on planar domains.

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.

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.

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.

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…

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 $D$. …

Sharp lower bound for first Neumann eigenvalue found in terms of diameter and width.

Our goal is to provide a novel method of representing 2D shapes, where each shape will be assigned a unique fingerprint - a computable approximation to a conformal map of the given shape to a canonical shape in 2D or 3D space (see page 22 for a few examples). In this paper, we make the first significant step in this pr…

We give a number of examples of isospectral pairs of plane domains, and a particularly simple method of proving isospectrality. One of our examples is a pair of domains that are not only isospectral but homophonic: Each domain has a distinguished point such that corresponding normalized Dirichlet eigenfunctions take eq…

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.

We develop a theory of planar, origin-symmetric, convex domains that are inextensible with respect to lattice covering, that is, domains such that augmenting them in any way allows fewer domains to cover the same area. We show that origin-symmetric inextensible domains are exactly the origin-symmetric convex domains wi…

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…

The paper constructs $λ$-hypersurfaces for $λ>0$ and $λ<0$.

This paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. In [CM3]-[CM5] we describe the case where the surfaces are topologically disks on any fixed small scale. To describe general planar domains (in [CM6]) we need in …

We study several quantities associated to the Green's function of a multiply connected domain in the complex plane. Among them are some intrinsic properties such as geodesics, curvature, and $L^2$-cohomology of the capacity metric and critical points of the Green's function. The principal idea used is an affine scaling…

In this paper, we provide new discrete uniformization theorems for bounded, $m$-connected planar domains. To this end, we consider a planar, bounded, $m$-connected domain $Ω$ and let $\bordΩ$ be its boundary. Let $\mathcal{T}$ denote a triangulation of $Ω\cup\bordΩ$. We construct a \emph{new} decomposition of $Ω\cup\bo…

New method uses contours of segmented images for X-ray classification.

The classical inverse problem of recovering a simply connected smooth planar domain from the Steklov spectrum \cite{E} is equivalent to the problem of recovering, up to a conformal equivalence, a positive function $a\in C^\infty({\mathbb S})$ on the unit circle ${\mathbb S}=\{e^{iθ}\}$ from the eigenvalue spectrum of t…

We generalize the familiar notions of overtwistedness and Giroux torsion in 3-dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order $k \ge 0$ can be interpreted as measuring a gradation in "degrees of tightness" of contact manifolds…

Selects points from Jordan domains on Riemannian surfaces.

In the present paper several bounds on multiplicities of eigenvalues of the Laplacian operator on surfaces are generalized from the case of either closed surface or simply-connected planar domain to the case of a surface of positive genus with holes.

The paper resolves a problem about metric inequivalence and characterizes proper holomorphic maps.

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

Defines renormalised energies for singular harmonic maps into compact manifolds.

In the paper, we focus on the connectedness of planar self-affine sets $T(A,{\mathcal{D}})$ generated by an integer expanding matrix $A$ with $|\det (A)|=3$ and a collinear digit set ${\mathcal{D}}=\{0,1,b\}v$, where $b>1$ and $v\in {\mathbb{R}}^2$ such that $\{v, Av\}$ is linearly independent. We discuss the domain of…

Given a smooth simply connected planar domain, the area is bounded away from zero in terms of the maximal curvature alone. We show that in higher dimensions this is not true, and for a given maximal mean curvature we provide smooth embeddings of the ball with arbitrary small volume.

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

For $p\in (1,2]$ and a bounded, convex, nonempty, open set $Ω\subset\mathbb R^2$ let $μ_p(\barΩ,\cdot)$ be the $p$-capacitary curvature measure (generated by the closure $\barΩ$ of $Ω$) on the unit circle $\mathbb S^1$. This paper shows that such a problem of prescribing $μ_p$ on a planar convex domain: "Given a finite…

Near the end of his life, Bernhard Riemann made the marvelous discovery of a 1-parameter family $R_λ$, $λ\in (0,\infty)$, of periodic properly embedded minimal surfaces in $\mathbb{R}^3$ with the property that every horizontal plane intersects each of his examples in either a circle or a straight line. Furthermore, as …

Study visibility properties of Kobayashi distance on unbounded domains.

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 equi-affine invariants for convex domains with asymptotes.