arXiv research

New knots found with Seifert genus not matching minimal genus Seifert surfaces.

Survey of minimal genus problem progress.

Proves prime theta-curves for knots on minimal genus surfaces.

The paper develops a theory for free boundary minimal surfaces with genus at least one.

Research on the least complex surface in certain 4D shapes.

Investigates minimal genus of second homology classes in RAAGs, finding bounds and specific cases.

Roberts proved that a family of alternating, arborescent, prime knots each have at least $2^{2n-1}$ distinct minimal genus Seifert surfaces, where $n$ is the genus of the knot in question. We give a subfamily of these knots that have exactly this many minimal genus Seifert surfaces.

Constructs a family of genus three minimal surfaces with parallel ends.

Given a one-dimensional homology class in a lens space, a question related to the Berge conjecture on lens space surgeries is to determine all knots realizing the minimal rational genus of all knots in this homology class. It is known that simple knots are rational genus minimizers. In this paper, we construct many non…

Complete minimal surfaces with any genus found in a specific 3-manifold.

Minimal genus surfaces solve homology problems in finite complexes.

Two non-diffeomorphic minimal genus trisections found for a 4-manifold.

The paper reconfirms a lower bound on rational genus using Heegaard Floer homology.

The paper explores the genus of surfaces in complex projective spaces and improves minimal genus bounds.

Minimal surfaces in spheres found for any genus.

The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.

Minimal crossing virtual links have minimal supporting genus.

We solve a certain case of the minimal genus problem for embedded surfaces in elliptic 4-manifolds. The proofs involve a restricted transitivity property of the action of the orientation preserving diffeomorphism group on the second homology. In the case we consider we get the minimal possible genus allowed by the adju…

We show that a 3-manifold containing an incompressible surface has topologically minimal surfaces of arbitrary high genus.

Unique genus 2 minimal surface in S^3 with specific symmetry group.

There exists a properly embedded minimal surface of genus one with one end. The end is asymptotic to the end of the helicoid. This genus one helicoid is constructed as the limit of a continuous one-parameter family of screw-motion invariant minimal surfaces--also asymptotic to the helicoid--that have genus equal to one…

Minimal singular fibers found in nonorientable Lefschetz fibrations.

The paper constructs surfaces of high genus with three ends.

The study finds a way to create minimal surfaces with specific shapes in 3-manifolds.

For every genus $g$, we prove that $S^2 \times R$ contains complete, properly embedded, genus-$g$ minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the $S^2$ tends to infinity, these examples converge smoothly to complete, properly embedded minimal s…

New minimal surfaces found in 3D sphere with low genus.

For any positive integer $g$, we completely determine the minimal genus function for $Σ_{g}\times T^{2}$. We show that the lower bound given by the adjunction inequality is not sharp for some class in $H_{2}(Σ_{g}\times T^{2})$. However, we construct a suitable embedded surface for each class and we have exact values o…

Paper finds minimal number of curves in surface systems.

For every genus g, we prove that S^2 x R contains complete, properly embedded, genus-g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the S^2 tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in Eu…

We consider the hyperelliptic handlebody group on a closed surface of genus $g$. This is the subgroup of the mapping class group on a closed surface of genus $g$ consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the …

Minimal maps from surfaces to torus found for various genus values.

This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.

The Weierstrass representation for minimal surfaces in $\mathbb{R}^3$ provides a flexible method for constructing minimal surfaces of arbitrary genus. The topological limitations of minimal surfaces interfere with this providing a more general geometric modeling tool. Minimal surfaces lie in the larger class of harmoni…

We prove optimal genus bounds for minimal surfaces arising from the min-max construction of Simon-Smith. This confirms a conjecture made by Pitts-Rubinstein in 1986.

We obtain new lower bounds of the minimal genus of a locally flat surface representing a 2-dimensional homology class in a topological 4-manifold with boundary, using the von Neumann-Cheeger-Gromov $ρ$-invariant. As an application our results are employed to investigate the slice genus of knots. We illustrate examples …

We use Colding--Minicozzi lamination theory to study the systole of large genus minimal surfaces in an ambient three-manifold of positive Ricci curvature.

Minimal surfaces in a ball have limited area.

Minimal tori that are linearly full in the 3-sphere possess a natural invariant g called their spectral genus, which was introduced by Hitchin. We show that for each g>0, there are countably many real g-dimensional families of minimally immersed tori with spectral genus g (two of these dimensions are just reparametrisa…

Proves existence of minimal surfaces of arbitrary genus with two ends.

An estimate for the genus function in circle bundles over irreducible 3-manifolds is proven. This estimate is in many cases an equality and it relates the minimal genus of the surfaces representing a given homology class with the self-intersection of the class and the Thurston norm of the underlying 3-manifold.

Given an element in the first homology of a rational homology 3-sphere $Y$, one can consider the minimal rational genus of all knots in this homology class. This defines a function $Θ$ on $H_1(Y;\mathbb Z)$, which was introduced by Turaev as an analogue of Thurston norm. We will give a lower bound for this function usi…

We employ min-max techniques to show that the unit ball in $\mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.

Characterizes area-minimizing maps for surfaces of genus ≥ 2.

Using Traizet's regeneration method, we prove that for each positive integer n there is a family of embedded, doubly periodic minimal surfaces with parallel ends in Euclidean space of genus 2n-1 and 4 ends in the quotient by the maximal group of translations. The genus 2n-1 family converges smoothly to 2n copies of Sch…

We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed.

It was shown by Usher that any fiber sum of Lefschetz fibrations over $S^2$ is minimal, which was conjectured by Stipsicz. We prove that the converse does not hold by showing that there exists an indecomposable minimal genus-2 Lefschetz fibration.

New maxfaces with Enneper ends found.

Researchers refine the non-orientable $4$-genus of torus knots using Batson's surfaces.