Number Of Boundary Components Research Articles

Bounds for the Morse index of free boundary minimal surfaces

Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the energy index and the area index of a free-boundary minimal surface of a Riemannian manifold with boundary, and show that the area index is controlled from above by the area and the topology of the surface. Combining these results with work of Fraser-Li, we conclude that the area index of a free-boundary minimal surface in a convex domain of Euclidean three-space, is bounded from above by a linear function of its genus and its number of boundary components. We also prove index bounds for submanifolds of higher dimension.

Degenerating sequences of conformal classes and the conformal Steklov spectrum

AbstractLet $\Sigma $ be a compact surface with boundary. For a given conformal class c on $\Sigma $ the functional $\sigma _k^*(\Sigma ,c)$ is defined as the supremum of the kth normalized Steklov eigenvalue over all metrics in c. We consider the behavior of this functional on the moduli space of conformal classes on $\Sigma $ . A precise formula for the limit of $\sigma _k^*(\Sigma ,c_n)$ when the sequence $\{c_n\}$ degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander–Nadirashvili invariants of closed manifolds defined as $\inf _{c}\sigma _k^*(\Sigma ,c)$ , where the infimum is taken over all conformal classes c on $\Sigma $ . We show that these quantities are equal to $2\pi k$ for any surface with boundary. As an application of our techniques we obtain new estimates on the kth normalized Steklov eigenvalue of a nonorientable surface in terms of its genus and the number of boundary components.

Lens spaces which are realizable as closures of homology cobordisms over planar surfaces

We determine the condition on a given lens space having a realization as a closure of homology cobordism over a planar surface with a given number of boundary components. As a corollary, we see that every lens space is represented as a closure of homology cobordism over a planar surface with three boundary components. In the proof of this corollary, we use Chebotarev’s density theorem.

Index estimates for free boundary constant mean curvature surfaces

In this paper, we consider compact free boundary constant mean curvature surfaces immersed in a mean convex body of the Euclidean space or in the unit sphere. We prove that the Morse index is bounded from below by a linear function of the genus and number of boundary components.

Extremal embedded graphs

Let G be a ribbon graph and μ( G ) be the number of components of the virtual link formed from G as a cellularly embedded graph via the medial construction. In this paper we first prove that μ( G ) ≤ f ( G ) + γ( G ) , where f ( G ) and γ( G ) are the number of boundary components and Euler genus of G , respectively. A ribbon graph is said to be extremal if μ( G ) = f ( G ) + γ( G ) . We then obtain that a ribbon graph is extremal if and only if its Petrial is plane. We introduce a notion of extremal minor and provide an excluded extremal minor characterization for extremal ribbon graphs. We also point out that a related result in the monograph by Ellis-Monaghan and Moffatt is not correct and prove that two related conjectures raised by Huggett and Tawfik hold for more general ribbon graphs.

Knot exteriors with all possible meridional essential surfaces

We show the existence of infinitely many knot exteriors where each of which contains meridional essential surfaces of any genus and (even) number of boundary components. That is, the compact surfaces that have a meridional essential embedding into a knot exterior have a meridional essential embedding into each of these knot exteriors. From these results, we also prove the existence of a hyperbolic knot exterior in some 3-manifold for which there are meridional essential surfaces of independently unbounded genus and number of boundary components.

Bordered surfaces in the 3-sphere with maximum symmetry

We consider orientation-preserving actions of finite groups G on pairs (S3,Σ), where Σ denotes a compact connected surface embedded in S3. In a previous paper, we considered the case of closed, necessarily orientable surfaces, determined for each genus g>1 the maximum order of such a G for all embeddings of a surface of genus g, and classified the corresponding embeddings.In the present paper we obtain analogous results for the case of bordered surfaces Σ (i.e. with non-empty boundary, orientable or not). Now the genus g gets replaced by the algebraic genus α of Σ (the rank of its free fundamental group); for each α>1 we determine the maximum order mα of an action of G, classify the topological types of the corresponding surfaces (topological genus, number of boundary components, orientability) and their embeddings into S3. For example, the maximal possibility 12(α−1) is obtained for the finitely many values α=2,3,4,5,9,11,25,97,121 and 241.

Index estimates for free boundary minimal hypersurfaces

We show that the Morse index of a properly embedded free boundary minimal hypersurface in a strictly mean convex domain of the Euclidean space grows linearly with the dimension of its first relative homology group (which is at least as big as the number of its boundary components, minus one). In ambient dimension three, this implies a lower bound for the index of a free boundary minimal surface which is linear both with respect to the genus and the number of boundary components. Thereby, the compactness theorem by Fraser and Li implies a strong compactness theorem for the space of free boundary minimal surfaces with uniformly bounded Morse index inside a convex domain. Our estimates also imply that the examples constructed, in the unit ball, by Fraser–Schoen and Folha–Pacard–Zolotareva have arbitrarily large index. Extensions of our results to more general settings (including various classes of positively curved Riemannian manifolds and other convexity assumptions) are discussed.

A lower bound of the distortion of the Torelli group in the mapping class group with boundary components

We prove that the Torelli group of an oriented surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus–Farb–Putman’s techniques. Further we show that the distortion of the Torelli group in the level [Formula: see text] mapping class group is the same as that in the mapping class group.

Index bounds for free boundary minimal surfaces of convex bodies

In this paper, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In particular, we show that the index of a free boundary minimal surface in a convex domain in $\mathbb{R}^3$ tends to infinity as its genus or the number of boundary components tends to infinity.

Automorphisms groups of simplicial complexes of infinite type surfaces

Let S be an orientable surface of innite genus with a nite numberof boundary components. In this work we consider the curve complex C(S), the nonseparating curve complex N(S), and the Schmutz graph G(S) of S. When all topological ends of S carry genus, we show that all elements in the automorphismgroups Aut(C(S)), Aut(N(S)), and Aut(G(S)) are geometric, i.e. these groups are naturally isomorphic to the extended mapping class group MCG(S) of the innite surface S. Finally, we study rigidity phenomena within Aut(C(S)) and Aut(N(S)).

Fillings of Genus–1 Open Books and 4–Braids

We show that there are contact 3-manifolds of support genus one which admit infinitely many Stein fillings, but do not admit arbitrarily large ones. These Stein fillings arise from genus-1 allowable Lefschetz fibrations with distinct homology groups, all filling a fixed minimal genus open book supporting the boundary contact 3-manifold. In contrast, we observe that there are only finitely many possibilities for the homology groups of Stein fillings of a given contact 3-manifold with support genus zero. We also show that there are 4-strand braids which admit infinitely many distinct Hurwitz classes of quasipositive factorizations, yielding in particular an infinite family of knotted complex analytic annuli in the 4-ball bounding the same transverse link up to transverse isotopy. These realize the smallest possible examples in terms of the number of boundary components a genus-1 mapping class and the number of strands a braid can have with infinitely many positive/quasipositive factorizations.

Erratum to: “On unique determination of 3-connected plane domains by relative conformal moduli of pairs of boundary components”

The boundary values of conformal mappings of plane finitely connected domains are studied. A complete description of the boundary values of such mappings in terms of the moduli (extremal lengths) of pairs of boundary components of domains is obtained in the case where the number of boundary components is not larger than three.

Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture

Abstract Let E ⊆ ℙ 2 {E\subseteq\mathbb{P}^{2}} be a complex rational cuspidal curve and let ( X , D ) → ( ℙ 2 , E ) {(X,D)\to(\mathbb{P}^{2},E)} be the minimal log resolution of singularities. We prove that E has at most six cusps and we establish an effective version of the Zaidenberg finiteness conjecture (1994) concerning Eisenbud–Neumann diagrams of E. This is done by analyzing the Minimal Model Program run for the pair ( X , 1 2 ⁢ D ) {(X,\frac{1}{2}D)} . Namely, we show that ℙ 2 ∖ E {\mathbb{P}^{2}\setminus E} is ℂ * * {\mathbb{C}^{**}} -fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.

Derived invariants for surface algebras

In this paper we study the derived equivalences between surface algebras, introduced by David-Roesler and Schiffler [11]. Each surface algebra arises from a cut of an ideal triangulation of an unpunctured marked Riemann surface with boundary. A cut can be regarded as a grading on the Jacobian algebra of the quiver with potential (Q,W) associated with the triangulation.Fixing a set ϵ of generators of the fundamental group of the surface π1(S), we associate to any cut d a weight wϵ(d)∈Z2g+b, where g is the genus of S and b the number of boundary components. The main result of the paper asserts that the derived equivalence class of the surface algebra is determined by the corresponding weight wϵ(d) up to homeomorphism of the surface. Surface algebras are gentle and of global dimension ≤2, and any surface algebras coming from the same surface (S,M) are cluster equivalent, in the sense of [2]. To prove that the weight is a derived invariant we strongly use results about cluster equivalent algebras from [2].Furthermore we also show that for surface algebras the invariant defined for gentle algebras by Avella-Alaminos and Geiss in [6] is determined by the weight.

Sharp eigenvalue bounds and minimal surfaces in the ball

We prove existence and regularity of metrics on a surface with boundary which maximize sigma_1 L where sigma_1 is the first nonzero Steklov eigenvalue and L the boundary length. We show that such metrics arise as the induced metrics on free boundary minimal surfaces in the unit ball B^n for some n. In the case of the annulus we prove that the unique solution to this problem is the induced metric on the critical catenoid, the unique free boundary surface of revolution in B^3. We also show that the unique solution on the Mobius band is achieved by an explicit S^1 invariant embedding in B^4 as a free boundary surface, the critical Mobius band. For oriented surfaces of genus 0 with arbitrarily many boundary components we prove the existence of maximizers which are given by minimal embeddings in B^3. We characterize the limit as the number of boundary components tends to infinity to give the asymptotically sharp upper bound of 4pi. We also prove multiplicity bounds on sigma_1 in terms of the topology, and we give a lower bound on the Morse index for the area functional for free boundary surfaces in the ball.

Minimal immersions of compact bordered Riemann surfaces with free boundary

Let N N be a complete, homogeneously regular Riemannian manifold of dim N ≥ 3 N \geq 3 and let M M be a compact submanifold of N N . Let Σ \Sigma be a compact orientable surface with boundary. We show that for any continuous f : ( Σ , ∂ Σ ) → ( N , M ) f: \left ( \Sigma , \partial \Sigma \right ) \rightarrow \left ( N, M \right ) for which the induced homomorphism f ∗ f_{*} on certain fundamental groups is injective, there exists a branched minimal immersion of Σ \Sigma solving the free boundary problem ( Σ , ∂ Σ ) → ( N , M ) \left ( \Sigma , \partial \Sigma \right ) \rightarrow \left ( N, M \right ) , and minimizing area among all maps which induce the same action on the fundamental groups as f f . Furthermore, under certain nonnegativity assumptions on the curvature of a 3 3 -manifold N N and convexity assumptions on the boundary M = ∂ N M=\partial N , we derive bounds on the genus, number of boundary components and area of any compact two-sided minimal surface solving the free boundary problem with low index.

Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

We prove two explicit bounds for the multiplicities of Steklov eigenvalues σ κ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index κ of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given Riemannian surface with smooth boundary the multiplicities of Steklov eigenvalues σκ are uniformly bounded in κ.

Link invariants via counting surfaces

A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain combinatorial types. These formulas generalize the calculation of a linking number by counting signs of crossings in a link diagram. Until recently, explicit formulas of this type were known only for few invariants of low degrees. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram. We then identify the resulting invariants with certain derivatives of the HOMFLYPT polynomial.

Topological insulators avoid the parity anomaly

The surface of a 3+1d-topological insulator hosts an odd number of gapless Dirac fermions when charge conjugation and time-reversal symmetries are preserved. Viewed as a purely 2+1d system, this surface theory would necessarily explicitly break parity and time-reversal when coupled to a fluctuating gauge field. Here, we explain why such a state can exist on the boundary of a 3+1d system without breaking these symmetries, even if the number of boundary components is odd. This is accomplished from two complementary perspectives: topological quantization conditions and regularization. We first discuss the conditions under which (continuous) large gauge transformations may exist when the theory lives on a boundary of a higher-dimensional space-time. Next, we show how the higher-dimensional bulk theory is essential in providing a parity-invariant regularization of the theory living on the lower-dimensional boundary or defect.