arXiv research

New inequalities for planar convex domains' Laplacian eigenvalues.

Hot spots conjecture proven for small eigenvalue domains.

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

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.

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

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…

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

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

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…

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

Study equi-affine invariants for convex domains with asymptotes.

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…

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…

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

Compact Special Weingarten surfaces with planar convex boundaries are disks.

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

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

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…

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

Selects points from Jordan domains on Riemannian surfaces.

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.

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…

The paper finds new inequalities for convex polygons.

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.

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

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…

New heat trace coefficients reveal curvature effects in polygonal domains.

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

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 wave trace of certain convex domains can be smooth near some points in the length spectrum.

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

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

Upper bounds for magnetic Laplacian eigenvalues on 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…

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

Let $Ω\subset R^2$ be a bounded piecewise smooth domain and $φ_λ$ be a Neumann (or Dirichlet) eigenfunction with eigenvalue $λ^2$ and nodal set ${ N}_{φ_λ} = {x \in Ω; φ_λ(x) = 0}.$ Let $H \subset Ω$ be an interior $C^ω$ curve. Consider the intersection number $n(λ,H):= \# (H \cap N_{φ_λ} ).$ We first prove that fo…

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…

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.

Study three discrete envelope types of polygon bisection lines.

Proves Birkhoff-Poritsky conjecture for centrally-symmetric billiards.

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.

The paper finds conditions for graphs with constant mean curvature in hyperbolic space.

In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, $p$-flow, for $1\leq p<\infty.$ Here we investigate the asymptotic behavior of the planar $p$-flow for $p=\infty$ in the class of smooth, origin-symme…