arXiv research

Describes 3-manifolds by families of singularly fibered surfaces.

Given two univalent harmonic mappings $f_1$ and $f_2$ on $\mathbb{D}$, which lift to minimal surfaces via the Weierstrass-Enneper representation theorem, we give necessary and sufficient conditions for $f_3=(1-s)f_1+sf_2$ to lift to a minimal surface for $s\in[0,1]$. We then construct such mappings from Enneper's surfa…

Defines connections for singularly foliated bundles.

In this paper we present a new family of solutions to the singularly perturbed Allen-Cahn equation $α^2 Δu + u(1-u^2)=0, \quad \hbox{in }Ω\subset \R^N$ where $N=3$, $Ω$ is a smooth bounded domain and $\A>0$ is a small parameter. We provide asymptotic behavior which shows that, as $α\to 0$, the level sets of the soluti…

Extends Frohman and Rannard's result to Seifert fiber spaces with singular surfaces.

Study complex surfaces fibered over Teichmüller curves with Veech fibers.

Constructs a family to handle unstable fibers on complex surfaces.

Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and o…

We show that a compact complex surface which fibers smoothly over a curve of genus >1 with fibers of genus >1 fibers holomorphically. We deduce an improvement of a result in [D Kotschick, Math. Research Letters, 5 (1998) 227-234], and a characterisation of fibered surfaces with zero signature.

Study shows quasipositive fiber surfaces can't be well-ordered.

Unique incompressible surface in fibered 3-manifolds.

Paper proves unbounded stabilization heights of fiber surfaces using Hopf invariant.

New research finds infinitely many hyperbolic knots not almost-fibered.

We define the surface complex for $3$-manifolds and embark on a case study in the arena of Seifert fibered spaces. The base orbifold of a Seifert fibered space captures some of the topology of the Seifert fibered space, so, not surprisingly, the surface complex of a Seifert fibered space always contains a subcomplex is…

Study on surfaces in flag threefold with constraints on twistor fibers.

New bounds found for minimal surfaces in hyperbolic 3-manifolds.

In this paper we give the first example of a surface bundle over a surface with at least three fiberings. In fact, for each $n \ge 3$ we construct $4$-manifolds $E$ admitting at least $n$ distinct fiberings $p_i: E \to Σ_{g_i}$ as a surface bundle over a surface with base and fiber both closed surfaces of negative Eule…

Study connects surface projections in fibered 3-manifolds.

We study principal curvatures of fibers and Heegaard surfaces smoothly embedded in hyperbolic 3-manifolds. It is well known that a fiber or a Heegaard surface in a hyperbolic 3-manifold cannot have principal curvatures everywhere less than one in absolute value. We show that given an upper bound on the genus of a minim…

Generalizes pseudo-product structures with abnormal extremals.

The study bounds and characterizes surfaces containing smooth conics and twistor fibers in a flag threefold.

We characterize cutting arcs on fiber surfaces that produce new fiber surfaces, and the changes in monodromy resulting from such cuts. As a corollary, we characterize band surgeries between fibered links and introduce an operation called Generalized Hopf banding. We further characterize generalized crossing changes bet…

For a closed manifold $M$, let Fib$(M)$ be the number of distinct fiberings of $M$ as a fiber bundle with fiber a closed surface. In this paper we give the first computation of Fib$(M)$ where $1<\text{Fib}(M)<\infty$ but $M$ is not a product. In particular, we prove Fib$(M)=2$ for the Atiyah-Kodaira manifold and any fi…

Classifies tight contact structures on specific Seifert fibered manifolds.

Study ribbon concordance and minimal compressions, proving new results about fibered knots.

Every Kauffman state σof a link diagram D(K) naturally defines a state surface S_σwhose boundary is K. For a homogeneous state σ, we show that K is a fibered link with fiber surface S_σif and only if an associated graph G'_σis a tree. As a corollary, it follows that for an adequate knot or link, the second and next-to-…

The paper studies mapping class groups of nontrivial $S^2$ fiber bundles.

In "Tunnel one, fibered links", the second author showed that the tunnel of a tunnel number one, fibered link can be isotoped to lie as a properly embedded arc in the fiber surface of the link. In this paper, we analyze how the arc behaves under the monodromy action, and show that the tunnel arc is nearly clean, with t…

In this paper, we first classify singular fibers of proper $C^\infty$ stable maps of 3-dimensional manifolds with boundary into surfaces. Then, we compute the cohomology groups of the associated universal complex of singular fibers, and obtain certain cobordism invariants for Morse functions on compact surfaces with bo…

Minimal singular fibers found in nonorientable Lefschetz fibrations.

For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface bundles over genus h surfaces whose monodromies are in the Torelli group (provided…

Study finds negatively curved spheres in elliptic surfaces and their modifications.

We prove that a Lefschetz fibration over the disc that, after compactification, has the same singular fibers as an extremal rational elliptic surface can be obtained by deleting a singular fiber and a section from the rational extremal elliptic surface, i.e. such a Lefschetz fibration is determined up to topological eq…

Study handles in sutured manifolds and knots, finding varied handle numbers and unique surfaces.

The study explores how surface diffeomorphisms of knots relate to their topological properties.

It is known that there are surface bundles of arbitrarily high genus which have genus two Heegaard splittings. The simplest examples are Seifert fibered spaces with the sphere as a base space, three exceptional fibers and which allow horizontal surfaces. We characterize the monodromy maps of all surface bundles with ge…

Resolves singular fibers of a 5-manifold with $S^1$ action.

The paper studies elliptic surfaces and proves unique fibered structures.

Proof of existence for ideal triangulations that normalize fibers in certain 3-manifolds.

Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classified by Floyd and Hatcher. We conver…

In this work, we prove the existence of a family of solutions of the Allen-Cahn equation with nonlinear Neumann boundary condition under some constraints, whose nodal sets concentrate asymptotically to a given volume nondegenerate capillary hypersurface in a compact Riemannian manifold. Our construction is inspired by …

Consider a dihedral cover $f: Y\to X$ with $X$ and $Y$ four-manifolds and $f$ branched along an oriented surface embedded in $X$ with isolated cone singularities. We prove that only a slice knot can arise as the unique singularity on an irregular dihedral cover $f: Y\to S^4$ if $Y$ is homotopy equivalent to $\mathbb{CP…

Study of Einstein structures for surface group representations in specific Lie groups.

Study Stein and Milnor fillings of links from surface singularities.

New groups found that don't virtually algebraically fiber, related to mapping class group orbits.

There exists a (relatively minimal) genus g Lefschetz fibration with only one singular fiber over a closed (Riemann) surface of genus h iff g>2 and h>1. The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of tw…

The E_8-manifold has several natural framed link descriptions, and we give an efficient method (via `grapes') for showing that they are indeed the same 4-manifold. This leads to explicit handle pictures for the perturbation of singular fibers in an elliptic surface to a collection of fishtails. In the same vein, we sho…

The paper constructs moduli spaces for genus one fibered K3 surfaces.