Related papers: Translation Surfaces arising from Right Regular Pr…

We study smooth, proper embeddings of noncompact surfaces in 4-manifolds, focusing on exotic planes and annuli, i.e., embeddings pairwise homeomorphic to the standard embeddings of R^2 and R^2-int D^2 in R^4. We encounter two uncountable…

We classify $n$-dimensional geometric graph manifolds with nonnegative scalar curvature, and first show that if $n>3$, the universal cover splits off a codimension 3 Euclidean factor. We then proceed with the classification of the…

We prove non-existence of nontrivial uniformly subsonic inviscid irrotational flows around several classes of solid bodies with two protruding corners, in particular vertical and angled flat plates; horizontal plates are the only case where…

The Veech group of a translation surface is the group of Jacobians of orientation-preserving affine automorphisms of the surface. We present an algorithm which constructs all translation surfaces with a given lattice Veech group in any…

While it is well known from examples that no interesting `halfspace theorem' holds for properly immersed complete $n$-dimensional self-translating mean curvature flow solitons in Euclidean space $\mathbb{R}^{n+1}$, we show that they must…

In this article, we continue to study the geometry of bisections of certain rational elliptic surfaces. As an application, we give examples of Zariski N + 1-plets of degree 2N + 4 whose irreducible components are an irreducible quartic…

In this article, we propose a new approach for describing and understanding knots and links in a 3-manifold through the use of an embedded non-orientable surface. Specifically, we define a plat-like representation based on this…

We solve a problem on filling by Levi-flat hypersurfaces for a class of totally real 2-tori in a real 4-manifold with an almost complex structure tamed by an exact symplectic form. As an application we obtain a simple proof of Gromov's…

A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard…

A translation surface in the three-dimensional sphere $\mathbb{S}^3$ is a surface generated by the quaternionic product of two curves, called generating curves. In this paper, we present rigidity results for such surfaces. We introduce an…

In this paper, we prove that any mean curvature flow translator $\Sigma^2 \subset \mathbb{R}^3$ with finite total curvature and one end must be a plane. We also prove that if the translator $\Sigma$ has multiple ends, they are asymptotic to…

We construct the first examples of rationally convex surfaces in the complex plane with hyperbolic complex tangencies. In fact, we give two very different types of rationally convex surfaces: those that admit analytic fillings by…

We construct examples of ergodic vertical flows in periodic configurations of Eaton lenses of fixed radius. We achieve this by studying a family of infinite translation surfaces that are $\mathbb{Z}^2$-covers of slit tori. We show that the…

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper by the authors, the problem of…

In this paper we describe all rotation $H$-hypersurfaces in $H^n \times R$ and use them as barriers to prove existence and characterization of certain vertical $H$-graphs and to give symmetry and uniqueness results for compact…

Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles…

We prove the nonexistence of lattice tilings of $\mathbb{Z}^n$ by Lee spheres of radius $2$ for all dimensions $n\geq 3$. This implies that the Golomb-Welch conjecture is true when the common radius of the Lee spheres equals $2$ and…

We specify what is meant for a polytope to be reconstructible from its graph or dual graph. And we introduce the problem of class reconstructibility, i.e., the face lattice of the polytope can be determined from the (dual) graph within a…

Coverings of undirected graphs are used in distributed computing, and unfoldings of directed graphs in semantics of programs. We study these two notions from a graph theoretical point of view so as to highlight their similarities, as they…

We prove that if a translating soliton can be expressed as the sum of two curves and one of these curves is planar, then the other curve is also planar and consequently the surface must be a plane or a grim reaper. We also investigate…