arXiv research

The paper proves nonexistence and existence results for minimal surfaces in R^4.

New formulas for minimal surfaces with specific end conditions.

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…

New method constructs surfaces with constant mean curvature.

The paper proves the existence of maxfaces with multiple swallowtails and planar ends.

For any m > 0, we construct properly embedded minimal surfaces in H^2 x R with genus zero, infinitely many vertical planar ends and m limit ends. We also provide examples with an infinite countable number of limit ends. All these examples are vertical bi-graphs.

Study of minimal surfaces in 4D with specific ends.

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…

We prove that a minimal oriented stable annular end in H^2 x R whose asymptotic boundary is contained in two vertical lines has finite total curvature and converges to a vertical plane. Furthermore, if the end is embedded then it is a horizontal graph.

The spinor representation is developed and used to investigate minimal surfaces in ${\bfR}^3$ with embedded planar ends. The moduli spaces of planar-ended minimal spheres and real projective planes are determined, and new families of minimal tori and Klein bottles are given. These surfaces compactify in $S^3$ to yield …

We give a positive answer to M. Traizet's open question about the existence of complete embedded minimal surfaces with Scherk-ends without planar geodesics. In the singly periodic case, these examples get close to an extension of Traizet's result concerning asymmetric complete minimal submanifolds of Euclidean space wi…

New maxfaces with catenoid or planar ends constructed using node-opening technique.

We show the existence of 1-parameter families of non-periodic, complete, embedded minimal surfaces in euclidean space with infinitely many parallel planar ends. In particular we are able to produce finite genus examples and quasi-periodic examples of infinite genus.

In 3-dimensional Euclidean space, Scherk second surfaces are singly periodic embedded minimal surfaces with four planar ends. In this paper, we obtain a natural generalization of these minimal surfaces in any higher dimensional Euclidean space ${\R}^{n+1}$, for $n \geq 3$. More precisely, we show that there exist $(n-1…

We use contact fiber sums of open book decompositions to define an infinite hierarchy of filling obstructions for contact 3-manifolds, called planar k-torsion for nonnegative integers k, all of which cause the contact invariant in Embedded Contact Homology to vanish. Planar 0-torsion is equivalent to overtwistedness, w…

Defines complex for infinite-type surfaces with non-planar ends.

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…

We study the moduli space of null curves in Klein's quartic in the four-dimensional (complex) projective plane using methods developed by Robert Bryant. As a consequence, we show that minimal surfaces with $9$ embedded planar ends do not exist and formulate some conjectures about the previous moduli space.

Given k>=2, we construct a (2k-2)-parameter family of properly embedded minimal surfaces in H^2 x R invariant by a vertical translation T, called Saddle Towers, which have total intrinsic curvature 4 pi(1-k), genus zero and 2k vertical Scherk-type ends in the quotient by T. As limits of those Saddle Towers, we obtain J…

Spatial embeddings of planar graphs can have higher unknotting numbers than crossing numbers.

New minimal surfaces derived from helicoids.

Characterizes minor-minimal separating projective planar graphs and their generalizations.

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 …

Minimal moves for surfaces in 4D identified.

Study on planar graph braid groups' second homology.

The study extends Tutte's conflict graph concept to nonplanar graphs.

We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the cor…

A Euclidean minimal torus with planar ends gives rise to an immersed Willmore torus in the conformal 3--sphere $S^3=\R^3\cup \{\infty\}$. The class of Willmore tori obtained this way is given a spectral theoretic characterization as the class of Willmore tori with reducible spectral curve. A spectral curve of this type…

Given a finite or infinite planar graph all of whose faces have degree 4, we study embeddings in the plane in which all edges have length 1, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the existence of such an embedding, as well as a description of the set of all such emb…

In \cite{4} Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph $G$ a polynomial, denoted $[G]$, in three variables, $A$, $B$ and $a$, satisfies the skein relation: $$ [\psdiag{2}{6}{overcross}]=…

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…

Well-quasi-orders proved on embedded planar graphs.

Study classifies Halin graphs with positive curvature.

Compact Special Weingarten surfaces with planar convex boundaries are disks.

In this paper we study how to distinguish two embeddings of a finite collection of disjoint circles into the plane up to planar isotopy. We adopt the spirit of the approach by V. Turaev, Operator Invariants of Tangles, Math. USSR-Izv. 35 (1990), 411--444, by considering a category of planar tangles and representing it …

Minimal simplicial complexes in high dimensions always contain complex links.

Continuous epimorphisms between certain mapping class groups are induced by homeomorphisms.

Study infinite genus surfaces and Schottky groups for uniformization.

Characterizes graphs with leveled embeddings and introduces new graph invariants.

In this paper we describe a new deformation that connects minimal disks with planar ends with minimal disks with helicoidal ends. In this way, we are able to construct a family of complete minimal surfaces with helicoidal ends that contains the singly periodic genus one helicoid of Hoffman, Karcher and Wei.

We prove two results on the classification of trivial Legendrian embeddings $g: G \rightarrow (S^3,ξ_{std})$ of planar graphs. First, the oriented Legendrian ribbon $R_g$ and rotation invariant $\text{rot}_g$ are a complete set of invariants. Second, if $G$ is 3-connected or contains $K_4$ as a minor, then the unique t…

Solenoids are ``inverse limits'' of the circle, and the classical knot theory is the theory of tame embeddings of the circle into the 3-space. We give some general study, including certain classification results, of tame embeddings of solenoids into the 3-space as the ``inverse limits'' of the tame embeddings of the ci…

We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.

We give explicit deformations of embeddings of abstractly planar graphs that lie on the standard torus $T^2 \subset \mathbb{R}^3$ and that contain neither a nontrivial knot nor a nonsplit link into the plane. It follows that ravels do not embed on the torus. Our results provide general insight into properties of molecu…

We consider unbranched Willmore surfaces in the Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index - the number of linearly independent variational directions that locally decrease the Willmore energy.…

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…

Proves existence of non-planar minimal disks in ellipsoids.

In this note we construct a vase of catenoids - a symmetric immersed minimal surface with planar and catenoid ends.