New knots found with Seifert genus not matching minimal genus Seifert surfaces.
problem Discrepancy between Seifert genus and minimal genus Seifert surfaces.
method Constructed knots with specific genus and handle numbers to demonstrate the discrepancy.
result Found knots where Seifert genus is not realized by minimal genus Seifert surfaces.
Survey of minimal genus problem progress.
problem Finding the smallest genus surface in 3-manifolds.
method Survey and review of existing research.
result Summarizes recent advancements in the minimal genus problem.
Proves prime theta-curves for knots on minimal genus surfaces.
problem Prime knots and essential arcs on Seifert surfaces.
method Analyzes prime knot unions with essential arcs on minimal genus surfaces.
result Each prime knot union an essential arc on a minimal genus Seifert surface is a prime theta-curve.
The paper develops a theory for free boundary minimal surfaces with genus at least one.
problem Finding minimal surfaces with specific genus and boundary conditions.
method Using sweepouts of surfaces of genus g≥1 and m≥1 ideal boundary components, the paper constructs a min-max theory for free boundary minimal surfaces.
result The width for the area functional can be achieved by a bubble tree limit of branched genus g free boundary minimal surfaces with nodes.
Researchers find the minimal genus for a specific type of surface bundle.
problem Determining the minimal genus of ΣgimesT2 bundles. method Constructing embedded surfaces to find exact minimal genus values.
result Exact values of minimal genus functions for ΣgimesT2 bundles are determined. Research on the least complex surface in certain 4D shapes.
problem Finding the simplest surface in specific four-dimensional shapes.
method Analyzing smooth four-manifolds to determine minimal genus.
result Results on the minimal genus for studied four-manifolds.
Investigates minimal genus of second homology classes in RAAGs, finding bounds and specific cases.
problem Minimal genus of second homology classes in right angled Artin groups.
method Lower bounds, characterizations, and examples to show minimal genus.
result Minimal genus is half the rank for complete graphs, trees, and complete bipartite graphs, and can be realized by disjoint unions of tori.
Roberts proved that a family of alternating, arborescent, prime knots each have at least 22n−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.
problem Creating embedded doubly periodic minimal surfaces with specific topological properties.
method Constructs a one-parameter family of surfaces with given Weierstrass data and solves the period problem.
result Solves the two dimensional period problem for the constructed surfaces.
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…
Embedded surfaces in a ball have any genus and connected boundary.
problem Existence of embedded free boundary minimal surfaces with specific properties.
method Min-max techniques applied to the unit ball in R3. result Existence of embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
Complete minimal surfaces with any genus found in a specific 3-manifold.
problem Constructing minimal surfaces with arbitrary genus.
method Classical desingularization method.
result Complete embedded minimal surfaces of finite topology constructed.
Minimal genus surfaces solve homology problems in finite complexes.
problem Finding surface representatives of homology classes with minimal genus.
method Minimizing genus and Euler characteristic, analyzing surgeries and homotopy.
result Minimizers are homotopic to cellwise coverings, but problem is undecidable in general.
Two non-diffeomorphic minimal genus trisections found for a 4-manifold.
problem Existence of non-diffeomorphic minimal genus trisections of the same 4-manifold.
method Introduced a simple operation to create a trisection diagram from a relative trisection diagram.
result Existence of two non-diffeomorphic minimal genus trisections of the same (g,k;p,b)-type 4-manifold. The paper reconfirms a lower bound on rational genus using Heegaard Floer homology.
problem Lowering the rational genus of knots in rational homology 3-spheres.
method Using Heegaard Floer homology and the d-invariant. result Same lower bound and minimizers as Ni and the first author's results.
The paper explores the genus of surfaces in complex projective spaces and improves minimal genus bounds.
problem Investigating the genus of surfaces in complex projective spaces.
method Analyzing knots and torus knots in CP2 and CP2#CP2. result The CP2-genus of knots is unbounded, unlike its topological counterpart. Minimal surfaces in spheres found for any genus.
problem Finding minimal surfaces with arbitrary genus in 3-spheres.
method Topological structure analysis and embedding theorem.
result Every positive Ricci curvature 3-sphere contains a genus g surface.
The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.
problem Counting minimal surfaces in 3-manifolds with positive Ricci curvature.
method An enumerative min-max theorem linking surface counts to topological properties.
result Every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.
Minimal crossing virtual links have minimal supporting genus.
problem Understanding the relationship between the number of crossings and the genus of virtual links.
method Developed a new parity theory for virtual links to prove minimal crossing implies minimal genus.
result 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…
Disk complexes show 3-sphere surfaces are topologically minimal.
problem Understanding minimal surfaces in 3-sphere topology.
method Analyzing disk complexes of genus >1 Heegaard surfaces.
result Genus >1 Heegaard surfaces have minimal index 2g-1.
Unique genus 2 minimal surface in S^3 with specific symmetry group.
problem Finding minimal surfaces in S^3 with specific symmetries.
method Analyzing isometry groups of known minimal surfaces and proving uniqueness.
result ξ_{2,1} is the unique minimal surface of genus 2 with bidihedral symmetry.
We show that a 3-manifold containing an incompressible surface has topologically minimal surfaces of arbitrary high genus.
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.
problem Finding the minimal number of singular fibers in nonorientable Lefschetz fibrations.
method Analyzing admissible nonorientable genus g Lefschetz fibrations over orientable surfaces.
result Existence of one singular fiber if and only if g ≥ 4 and h ≥ 1.
The paper constructs surfaces of high genus with three ends.
problem Creating minimal surfaces of high genus with specific properties.
method One-parameter family of minimal surfaces constructed in Euclidean 3-space.
result The family includes the Costa-Hoffman-Meeks surfaces with two catenoidal ends and a flat middle end.
The study finds a way to create minimal surfaces with specific shapes in 3-manifolds.
problem Creating minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature.
method Develops a min-max theory and shows deformability of surfaces in a generic metric.
result Establishes a theorem for producing minimal surfaces with prescribed genus.
The study constructs minimal surfaces in a product space with specific properties.
problem Constructing minimal surfaces with specific topological and geometric properties in a product space.
method 1-parameter families of complete properly Alexandrov-embedded minimal surfaces with dihedral symmetry and finite total curvature.
result Examples of minimal surfaces with genus 1 and 2k ends in quotient spaces.
For every genus g, we prove that S2×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 S2 tends to infinity, these examples converge smoothly to complete, properly embedded minimal s…
New minimal surfaces found in 3D sphere with low genus.
problem Finding minimal surfaces in 3D sphere with low genus.
method Equivariant min-max procedure applied to a novel sweepout.
result Discovered new minimal surfaces with low genus.
Minimal hyperbolic surface diameter grows logarithmically with genus.
problem Finding the smallest possible diameter of hyperbolic surfaces.
method Random construction, lattice point counting, and exploration of random trivalent graphs.
result Minimal diameter is asymptotic to log(g) as genus g approaches infinity.
Paper finds minimal number of curves in surface systems.
problem Determining minimal number of curves in surface systems.
method Analyzes oriented surfaces of genus g for positive integers k and g.
result Exact minimal number of curves in filling k-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.
problem Finding minimal degree maps from genus g surfaces to the torus. method Constructing simplicial degree d maps from a triangulation of a genus g surface to the 7-vertex triangulation of the torus. result Minimal maps exist for g≥1 and ∣d∣≥2g−1 for g≥3. This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.
problem Classifying and determining the length of the shortest filling pairs on a specific type of surface.
method Classifying and determining the length of the shortest filling pairs on a specific type of surface.
result The paper classifies and determines the length of the shortest minimal filling pairs on a genus two surface.
The Weierstrass representation for minimal surfaces in R3 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.
problem Bounding the area of genus zero minimal surfaces in a unit ball.
method Proving an area inequality and showing convergence of saturating sequences.
result The area of each nonflat surface is less than its radial projection, with sharp asymptotic bounds.
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.
problem Proves existence of complete minimal surfaces of arbitrary genus with specific properties.
method Extends orthodisk method with partial symmetry to introduce controlled symmetry.
result Proves existence of Angel surfaces of arbitrary genus.
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 H1(Y;Z), which was introduced by Turaev as an analogue of Thurston norm. We will give a lower bound for this function usi…
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.
Characterizes area-minimizing maps for surfaces of genus ≥ 2.
problem Equivariant area-minimizing maps on surface covers.
method Classifies minimal surfaces in Hilbert spheres with constant negative Gaussian curvature.
result Characterizes all equivariantly area-minimizing maps from the universal cover of a surface to a Hilbert sphere.
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…
It was shown by Usher that any fiber sum of Lefschetz fibrations over S2 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.