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Let $X$ be a bundle over $S^1$ with fiber a 3--manifold $M$ and with monodromy $\varphi$. Gay and Kirby showed that if $\varphi$ fixes a genus $g$ Heegaard splitting of $M$ then $X$ has a genus $6g+1$ trisection. Genus $3g+1$ trisections have been found in certain special cases, such as the case where $\varphi$ is trivial, and it is known that trisections of genus lower than $3g+1$ cannot exist in general. We generalize these results to prove that there exists a trisection of genus $3g+1$ whenever $\varphi$ fixes a genus $g$ Heegaard surface of $M$. This means that $\varphi$ can be nontrivial, and can preserve or switch the two handlebodies of the Heegaard splitting. We additionally describe an algorithm to draw a diagram for such a trisection given a Heegaard diagram for $M$ and a description of $\varphi$.

We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichm\"{u}ller geodesic flow., Comment: 50 pages

We consider an embedded n-dimensional compact complex manifold in $$n+d$$ dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert’s formal principle program. We will give conditions ensuring that a neighborhood of $$C_n$$ in $$M_{n+d}$$ is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold’s result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in $$M_{n+d}$$ having $$C_n$$ as a compact leaf, extending Ueda’s theory to the high codimension case. Both problems appear as a kind of linearization problems involving small divisors condition arising from solutions to their cohomological equations.

This article examines the costumes of Claire Denis’s Chocolat. A highly aestheticized drama of pressed khaki uniforms and pith helmets in colonial Cameroon, clothing is, as this article argues, enfolded into a far broader topography of material power; it is an apparatus through which racialized and gendered difference is actively ‘fashioned’ for the screen. In Chocolat, it becomes clear that fashion and celluloid are intimately intertwined (‘sutured’ together, through Denis’s editing), with the bio-political containment of white femininity, historically underwritten by French society’s anxieties concerning racial miscegenation and sexual excesses in the colonies. Concurrently, however, dress also grants expression to transgressive currents of desire, which, in Denis’s provocative portrait of interracial attraction, intersect vividly with the sexual politics of fetishism. In the very fabrics of its material content – costumes, the pliable ‘skin’ of its images and the tissue of human relationships – Chocolat portrays colonial identity as deeply conflicted, and Denis affirms the material world as an agile and highly transgressive force in the theatre of colonial power.

In a text (1892) on light, Jules Henri Poincaré introduced a geometrical device for tracking the polarization of state of light interacting with matter. Poincaré first mapped all polarization ellipses onto the surface of sphere and changes of state were represented as rotations from one point to another about prescribed axes depending on linear optical properties of the medium. Here, we consider how Poincaré, the mathematician, invented his sphere. Poincaré's professional activities in the service of geodesy appear at first glance to provide a borrowed geometry for his one-to-one mapping of polarization ellipses to global lines of latitude and longitude. However, this association falls apart in the face of a close reading of Poincaré's biography and his influences, especially the research interests of his teachers from the École Polytechnique and the Écoles de Mines. The work of Tissot and Mallard on distortion ellipses in cartography and the etiology of optical activity in crystals, respectively, together with Poincaré's own study of the qualitative theory of differential equations, provide the iconography of, and motivation for, the sphere. Whether Poincaré's mentors were unwitting partners in the invention of the sphere-it is impossible to be sure-Poincaré's otherworldly geometric sensibilities carried him through isometries (rotations) on hyperbolic planes and beyond. The apparent ingenuity behind Poincaré's sphere is diminished in comparison to his fulsome achievements. Moreover, Poincaré's considerations of the psychology of invention further emphasize that sometimes great ideas arrive to those fortunate to receive them by mechanisms that resist interpretation.

The lines of curvature of a surface embedded in $\R^3$ comprise its principal foliations. Principal foliations of surfaces embedded in $\R^3$ resemble phase portraits of two dimensional vector fields, but there are significant differences in their geometry because principal foliations are not orientable. The Poincar\'e-Bendixson Theorem precludes flows on the two sphere $S^2$ with recurrent trajectories larger than a periodic orbit, but there are convex surfaces whose principal foliations are closely related to non-vanishing vector fields on the torus $T^2$. This paper investigates families of such surfaces that have dense lines of curvature at a Cantor set $C$ of parameters. It introduces discrete one dimensional return maps of a cross-section whose trajectories are the intersections of a line of curvature with the cross-section. The main result proved here is that the return map of a generic surface has \emph{breaks}; i.e., jump discontinuities of its derivative. Khanin and Vul discovered a qualitative difference between one parameter families of smooth diffeomorphisms of the circle and those with breaks: smooth families have positive Lebesgue measure sets of parameters with irrational rotation number and dense trajectories while families of diffeomorphisms with a single break do not. This paper discusses whether Lebesgue almost all parameters yield closed lines of curvature in families of embedded surfaces.

Let $X$ be an infinite Riemann surface equipped with its conformal hyperbolic metric such that the action of the covering group $\pi_1(X)$ on $\tilde{X}$ is of the first kind-i.e., the surface $X$ is equal to its convex core. We first prove that any geodesic lamination on $X$ is nowhere dense. Given a fixed geodesic pants decomposition of $X$ we define a family of train tracks on $X$ such that any geodesic lamination of $X$ is weakly carried by at least one train track. Then we parametrize all measured laminations on $X$ carried by a train track by the corresponding edge weight systems on the train track. Furthermore, we show that the weak* topology on the measured laminations weakly carried by a train track corresponds to a pointwise (weak) convergence of the edge weight systems. When one considers the Teichm\"uller space $T(X)$ of the Riemann surface $X$, it is natural to restrict the attention to the space $ML_b(X)$ of bounded measured laminations. When $X$ has a bounded geometry, we prove that a measured lamination weakly carried by a train track is bounded if and only if the corresponding edge weight system has a finite supremum norm. The Teichm\"uller space considerations lead to a natural uniform weak* topology on the space of bounded measured laminations on $X$. We prove that the correspondence between bounded measured laminations weakly carried by a train track and their edge weight systems is a homeomorphism when $ML_b(X)$ is equipped with the uniform weak* topology and the edge weight system is equipped with the topology induced by the supremum norm., Comment: 46 pages,10 figures

In 1994, Ittai Kan provided the first examples of maps with intermingled basins. The Kan example corresponds to a partially hyperbolic endomorphism defined on a surface with the boundary exhibiting two intermingled hyperbolic physical measures. Both measures are supported on the boundary, and they are also measures maximizing the topological entropy. In this work, we prove the existence of a third hyperbolic measure supported in the interior of the cylinder that maximizes the entropy for a larger class of maps including the Kan example. We also prove this statement for a larger class of invariant measures of large class maps including perturbations of the Kan example., Comment: Major modification including referees commentaries and improvement of the results were included in this version

We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David and Semmes showed that, given an Ahlfors $d$-regular set $E$, if we consider the set $\mathscr{B}$ of surface cubes (in the sense of Christ and David) near which $E$ does not look approximately like a union of planes, then $E$ is UR if and only if $\mathscr{B}$ satisfies a Carleson packing condition, that is, for any surface cube $R$, \[ \sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d} \lesssim ({\rm diam} R)^{d}.\] We show that, for lower content regular sets that aren't necessarily Ahlfors regular, if $\beta_{E}(R)$ denotes the square sum of $\beta$-numbers over subcubes of $R$ as in the Traveling Salesman Theorem for higher dimensional sets [AS18], then \[ \mathscr{H}^{d}(R)+\sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d}\sim \beta_{E}(R). \] We prove similar results for other uniform rectifiability critera, such as the Local Symmetry, Local Convexity, and Generalized Weak Exterior Convexity conditions. En route, we show how to construct a corona decomposition of any lower content regular set by Ahlfors regular sets, similar to the classical corona decomposition of UR sets by Lipschitz graphs developed by David and Semmes., Comment: 39 pages. To appear in Analysis & PDEs

This research explored the corrosion effect of NaCl solution on the surface quality of AZ31 magnesium alloy. Cubic-shaped specimens were immersed in standard 3.5% NaCl solution for 1, 2, 3, and 4 hours. The size of corrosion pits, fatigue crack propagation area, and surface topology were characterized. Fatigue tests were carried out on pre-corroded AZ31 specimens under different stress amplitudes. The correlation between the corrosion time and the virtual crack size was obtained. The virtual crack size is a new parameter that can relate the corrosion to the fatigue life. A new model was proposed to map the surface topology on the virtual crack size of pre-corroded AZ31 magnesium alloy. The fatigue life of 1-4-hour pre-corroded AZ31 specimens could be predicted accurately by this new model. The Levenberg-Marquardt and curve-fitting methods were used to obtain the constant parameters. The relative error between the crack size corresponding to the immersion time and the virtual crack size calculated by the proposed model was less than 1%.

We show that the action of the Kauffman bracket skein algebra of a surface $\Sigma$ on the skein module of the handlebody bounded by $\Sigma$ is faithful if and only if the quantum parameter is not a root of 1., Comment: 13 pages

The number $Z(n):=\lfloor n/2\rfloor\lfloor (n-1)/2\rfloor$ is the smallest number of crossings in a simple planar drawing of $K_{2,n}$ in which both vertices on the 2-side have the same clockwise rotation. For two vertices $u,v$ on the $q$-side of a simple drawing of $K_{p,q}$, let $\operatorname{cr}_D(u,v)$ denote the total number of crossings that edges incident with $u$ have with edges incident with $v$. We show that in any simple drawing $D$ of $K_{p,q}$ in a surface $\Sigma$ the number of pairs of vertices on the $q$-side of $K_{p,q}$ having $\operatorname{cr}_D(u,v)

The Cauchy problem for the well-known Benjamin–Bona–Mahoney–Burgers equation in the class of Holder initial functions from $${{\mathbb{C}}^{{2 + \alpha }}}({{\mathbb{R}}^{3}})$$ with $$\alpha \in (0,1]$$ is considered. For such initial functions, it is proved that the Cauchy problem has a unique time-unextendable classical solution in the class $${{\mathbb{C}}^{{(1)}}}([0,T];{{\mathbb{C}}^{{2 + \lambda }}}({{\mathbb{R}}^{3}}))$$ for any $$T \in (0,{{T}_{0}});$$ moreover, either $${{T}_{0}} = + \infty $$ or $${{T}_{0}} < + \infty $$ and, in the latter case, $${{T}_{0}}$$ is the solution blow-up time. To prove the solvability of the Cauchy problem, we examine volume and surface potentials associated with the Cauchy problem in Holder spaces. Finally, a Schauder estimate is obtained.

The paper proposes the first results of an ongoing research on the geometry of ribbed cross vaults starting from the principles and theories illustrated in some historical treatises. In the research, a particular role is assumed by algorithmic modelling which is an effective tool for the construction and verification of the geometry of the intrados surfaces of vaults. Starting from the studies conducted in this regard by the two Italian engineers. The construction of the digital models deriving from treatises by Curioni and Lamberti has allowed us to compare them with the geometry of the vaults of a side chapel of the church of Santa Lucia in Cagliari. Through the 3D laser scanner survey of these vaults, it was possible to evaluate which of the models proposed in the two treatises best approximated the case study of Santa Lucia in order to conduct algorithmic modelling of the surface closest to the geometry of the vaults revealed by the point cloud.

The theory of dessins d’enfants on compact Riemann surfaces, which are bipartite maps on compact orientable surfaces, are combinatorial objects used to study branched covers between compact Riemann surfaces and the absolute Galois group of the field of rational numbers. In this paper, we show how this theory is naturally extended to non-compact orientable surfaces and, in particular, we observe that the Loch Ness monster (LNM; the surface of infinite genus with exactly one end) admits infinitely many regular dessins d’enfants (either chiral or reflexive). In addition, we study different holomorphic structures on the LNM, which come from homology covers of compact Riemann surfaces, and infinite hyperelliptic and infinite superelliptic curves.

For any compact Riemannian surface of genus three (Σ,ds2) Yang and Yau proved that the product of the first eigenvalue of the Laplacian λ1(ds2) and the area Area(ds2) is bounded above by 24π. In this paper we improve the result and we show that λ1(ds2)Area(ds2)≤16(4−√7)π≈21.668π. About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value ≈21.414π., MINECO/FEDER grants no. MTM2017-89677-P, Junta Andalucía grants no. P06-FQM-01642 and P18-FR-4049

The generalization of McShane’s identity for the quotient surface $$\Sigma_q$$ of any Hecke group $$H_q$$ , where q is an integer greater than 3.

Using the local picture of the degeneration of sequences of minimal surfaces developed by Chodosh, Ketover and Maximo we show that in any closed Riemannian 3-manifold $(M,g)$, the genus of an embedded CMC surface can be bounded only in terms of its index and area, independently of the value of its mean curvature. We also show that if $M$ has finite fundamental group, the genus and area of any non-minimal embedded CMC surface can be bounded in term of its index and a lower bound for its mean curvature., 14 pages; v2: small improvements, replaced to reflect submission to The Journal of Geometric Analysis; v3: reflects version accepted by the J. Geom. Anal., main results are unchanged

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, \ldots . A natural toy model for further investigations is the M\"obius strip, a non-orientable surface with Euler characteristic $0$, and particularly the "square" M\"obius strip whose eigenvalues have higher multiplicities. In this case, we prove that the only Courant-sharp Dirichlet eigenvalues are the first and the second, and we exhibit peculiar nodal patterns., Comment: Revised version prior to publication. Accepted for publication in Portugaliae Mathematica

This study aims to estimate the entrance surface dose (ESD) of a water phantom for kilovoltage x-ray beams using an air kerma area product meter (KAP meter) equipped in an x-ray unit. The KAP meter was calibrated in terms of the ESD determined by a plane-parallel ionization chamber based on a 60Co absorbed dose-to-water calibration coefficient, ${N}_{D,w}^{{}^{60}\mathrm{C}\mathrm{o}}$. The ESD measured using the KAP meter was verified by comparing it with that estimated by the air kerma calibration coefficient, NK, for x-ray beam qualities. The ratio of ESDs based on ${N}_{D,w}^{{}^{60}\mathrm{C}\mathrm{o}}$ and NK was 1.003 on average and independent of the beam quality. The ESD by the KAP meter was an agreement within ±1.5% with that measured using the plane-parallel chamber for 10 × 10–30 × 30 cm2 fields with a source-surface distance of 75–150 cm. It was possible to estimate the ESD directly in a water phantom for x-ray beams without correction factors compared to the existing air kerma calibration, using a KAP meter calibrated based on ${N}_{D,w}^{{}^{60}\mathrm{C}\mathrm{o}}$.

We address the question of determining which mapping class groups of infinite-type surfaces admit nonelementary continuous actions on hyperbolic spaces. More precisely, let $\Sigma$ be a connected, orientable surface of infinite type with tame endspace whose mapping class group is generated by a coarsely bounded subset. We prove that $\mathrm{Map}(\Sigma)$ admits a continuous nonelementary action on a hyperbolic space if and only if $\Sigma$ contains a finite-type subsurface which intersects all its homeomorphic translates. When $\Sigma$ contains such a nondisplaceable subsurface $K$ of finite type, the hyperbolic space we build is constructed from the curve graphs of $K$ and its homeomorphic translates via a construction of Bestvina, Bromberg and Fujiwara. Our construction has several applications: first, the second bounded cohomology of $\mathrm{Map}(\Sigma)$ contains an embedded $\ell^1$; second, using work of Dahmani, Guirardel and Osin, we deduce that $\mathrm{Map}(\Sigma)$ contains nontrivial normal free subgroups (while it does not if $\Sigma$ has no nondisplaceable subsurface of finite type), has uncountably many quotients and is SQ-universal., Comment: v2: New title, updated references

We consider the analogue of the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke eigenvalues summed over quadratic progressions. Our reduction provides an analogue for the compact case of a criterion established by Luo--Sarnak for the case of the non-compact modular surface. The novelty is that known proofs of such criteria have depended crucially upon Fourier expansions, which are not available in the compact case. Unconditionally, we establish a twisted variant of the Holowinsky--Soundararajan theorem involving restrictions of normalized Hilbert modular forms arising via base change., Comment: 31 pages; to appear in IMRN

Let f be a harmonic map from a Riemann surface to a Riemannian n-manifold. We prove that if there is a holomorphic diffeomorphism h between open subsets of the surface such that $$f\circ h = f$$ , then f factors through a holomorphic map onto another Riemann surface. If such h is anti-holomorphic, we obtain an analogous statement. For minimal maps, this result is well known and is a consequence of the theory of branched immersions of surfaces due to Gulliver–Osserman–Royden. Our proof relies on various geometric properties of the Hopf differential.

In this article we obtain a rigid isotopy classification of generic pointed quartic curves (A, p) in $${\mathbb {R}}{\mathbb {P}}^{2}$$ R P 2 by studying the combinatorial properties of dessins. The dessins are real versions, proposed by Orevkov (Ann Fac Sci Toulouse 12(4):517–531, 2003), of Grothendieck’s dessins d’enfants. This classification contains 20 classes determined by the number of ovals of A, the parity of the oval containing the marked point p, the number of ovals that the tangent line $$T_p A$$ T p A intersects, the nature of connected components of $$A\setminus T_p A$$ A \ T p A adjacent to p, and in the maximal case, on the convexity of the position of the connected components of $$A\setminus T_p A$$ A \ T p A . We study the combinatorial properties and decompositions of dessins corresponding to real uninodal trigonal curves in real ruled surfaces. Uninodal dessins in any surface with non-empty boundary can be decomposed in blocks corresponding to cubic dessins in the disk $${\mathbf {D}}^2$$ D 2 , which produces a classification of these dessins. The classification of dessins under consideration leads to a rigid isotopy classification of generic pointed quartic curves in $${\mathbb {R}}{\mathbb {P}}^{2}$$ R P 2 . This classification was first obtained in Rieken (Geometr Ded 185(1):171–203, 2016) based on the relation between quartic curves and del Pezzo surfaces.

We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

In 1973, Brown, Erdős and Sós proved that if $\mathcal{H}$ is a 3-uniform hypergraph on $n$ vertices which contains no triangulation of the sphere, then $\mathcal{H}$ has $O(n^{5/2})$ edges, and this bound is the best possible up to a constant factor. Resolving a conjecture of Linial, also reiterated by Keevash, Long, Narayanan and Scott, we show that the same result holds for triangulations of the torus. Furthermore, we extend our result to every closed orientable surface $\mathcal{S}$.

For a punctured surface (Formula presented.), we characterize the representations of its fundamental group into (Formula presented.) that arise as the monodromy of a meromorphic projective structure on (Formula presented.) with poles of order at most two and no apparent singularities. This proves the analogue of a theorem of Gallo�Kapovich�Marden concerning (Formula presented.) -structures on closed surfaces, and settles a long-standing question about characterizing monodromy groups for the Schwarzian equation on punctured spheres. The proof involves a geometric interpretation of the Fock�Goncharov coordinates of the moduli space of framed (Formula presented.) -representations, following ideas of Thurston and some recent results of Allegretti�Bridgeland. © 2021 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

We show that the Torelli group of a closed surface of genus [Formula: see text] acts nontrivially on the rational cohomology of the space of [Formula: see text]-element subsets of that surface. This implies that for a Riemann surface of genus [Formula: see text], the mixed Hodge structure on the space of its positive, reduced divisors of degree [Formula: see text] does in general not split over [Formula: see text].

We show that the mapping class group of any closed connected orientable surface of genus at least five is generated by only two commutators, and if the genus is three or four, by three commutators.

In this paper, a generic procedure for the development and subsequent validation of the Riemann surface structure (RSS) for a punctured curved surface lying on a Riemann surface is discussed. The proposed procedure differs from the existing methods involving triangular meshes and rectangular grids that rely on induced patches on surfaces. This procedure can be applied to non-punctured surfaces as well as to surfaces with irregularly located punctures. Further, by defining appropriate transition functions, the proposed procedure eliminates the requirement for smooth transitions across the boundaries of adjacent patches. The analytic formulations of the RSS for an ellipsoid and a sphere are elaborated using the proposed procedure. Moreover, the RSS of a sphere defined through a family of conformal unit discs is proven equivalent to that defined by an existing method based on stereographic projection. This study proves that a smooth projection between the surface and (a subset of) the complex plane , can be remapped to the original surface.

The forbidding harshness of Galina Ustvolskaya's Sixth Piano Sonata can easily be associated with the spiritual and reductively statuesque imagery associated with this composer. Adapting and expanding an initial classification by Andreas Holzer and Tatiana Marković, this article proposes an in-depth analysis of loudness in Ustvolskaya's final piano composition. A detailed overview shows how she uses expanding cluster types, accentuation, and rising intensity to create prolonged dynamic arcs where at first glance dynamic progression appears to be largely flat. I go on to identify three moments of musical climax, all of which seem to articulate the golden ratio for an important subsection of the sonata. Based on these observations, I formulate two contrasting interpretations, one teleologically orientated, the other symmetrical, which show that the Sonata is governed by the tension between two distinct types of formal logic. My analysis uncovers a surprisingly multi-faceted structure, one that belies the austere image of "the composer with the hammer."

Ionic liquids (ILs), 1-alkyl-3-methylimidazolium alanine [Cnmim][Ala] (n = 2, 4), were synthesized and characterized by 1H NMR and DSC. Then, the average temperature vaporization enthalpy, $$ \Delta _\text{l}^{\text g} H_{\text m}^{\uptheta } $$ (Tav), of [Cnmim][Ala] (n = 2, 4) was determined by thermogravimetric method. The difference in heat capacities of ionic liquids between the vapor phase and the liquid phase, $$ \Delta _{\text l}^{g} C_{\text {p,m}}^{\uptheta } $$ , was calculated and $$ \Delta _\text{l}^{\text g} H_{\text m}^{\uptheta } $$ (Tav) was converted to $$ \Delta _\text{l}^{\text g} H_{\text m}^{\uptheta } $$ (298). In addition, the polarity, δμ, of two ILs was estimated based on the $$ \Delta _\text{l}^{\text g} H_{\text m}^{\uptheta } $$ (298). At the same time, the molar surface Gibbs free energy, gs, was calculated, and the traditional Eotvos equation was improved, and its parameters have clear physical significance. In addition, expression for predicting surface tension, γ, of the ILs is obtained by the combination gs with Lorentz–Lorenz equation, and the results show that both are in good agreement with the experimental values.

We consider the Birman-Hilden inclusion $\varphi\colon\mathfrak{Br}_{2g+1}\to\Gamma_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $\varphi$ is stably trivial in homology with twisted coefficients in the symplectic representation $H_1(\Sigma_{g,1})$ of the mapping class group; this generalises a result of Song and Tillmann regarding homology with constant coefficients. Furthermore we show that the stable homology of the braid group with coefficients in $\varphi^*(H_1(\Sigma_{g,1}))$ has only $4$-torsion., Comment: 17 pages, 3 figures, slight change in the introduction

Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n ∈ $\mathbb N$ for which there exists a surjection between the n- and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups.

This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ ( U , V ) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , ( λ , μ ) ↦ 𝔏 ⁢ ( λ , μ ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ ( U , V ) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

Let $S$ be a connected closed oriented surface of genus $g$. Given a triangulation (resp. quadrangulation) of $S$, define the index of each of its vertices to be the number of edges originating from this vertex minus $6$ (resp. minus $4$). Call the set of integers recording the non-zero indices the profile of the triangulation (resp. quadrangulation). If $\kappa$ is a profile for triangulations (resp. quadrangulations) of $S$, for any $m\in \mathbb{Z}_{>0}$, denote by $\mathscr{T}(\kappa,m)$ (resp. $\mathscr{Q}(\kappa,m)$) the set of (equivalence classes of) triangulations (resp. quadrangulations) with profile $\kappa$ which contain at most $m$ triangles (resp. squares). In this paper, we will show that if $\kappa$ is a profile for triangulations (resp. for quadrangulations) of $S$ such that none of the indices in $\kappa$ is divisible by $6$ (resp. by $4$), then $\mathscr{T}(\kappa,m)\sim c_3(\kappa)m^{2g+|\kappa|-2}$ (resp. $\mathscr{Q}(\kappa,m) \sim c_4(\kappa)m^{2g+|\kappa|-2}$), where $c_3(\kappa) \in \mathbb{Q}\cdot(\sqrt{3}\pi)^{2g+|\kappa|-2}$ and $c_4(\kappa)\in \mathbb{Q}\cdot\pi^{2g+|\kappa|-2}$. The key ingredient of the proof is a result of J. Kollar on the link between the curvature of the Hogde metric on vector subbundles of a variation of Hodge structure over algebraic varieties, and Chern classes of their extensions. By the same method, we also obtain the rationality (up to some power of $\pi$) of the Masur-Veech volume of arithmetic affine submanifolds of translation surfaces that are transverse to the kernel foliation.

In this paper, we study global bifurcation diagrams and the exact multiplicity of positive solutions for Minkowski-curvature problem $$\begin{aligned} \left\{ \begin{array}{l} -\left( u^{\prime }/\sqrt{1-{u^{\prime }}^{2}}\right) ^{\prime }=\lambda \exp u,\text { in }\left( -L,L\right) , \\ u(-L)=u(L)=0, \end{array} \right. \end{aligned}$$ where $$\lambda >0$$ is a bifurcation parameter and $$L>0$$ is an evolution parameter. It can be viewed as a variant of the one-dimensional Liouville–Bratu–Gelfand problem. We prove that there exists $$L_{0}>0$$ such that the bifurcation curve $$S_{L}$$ is S-shaped for $$L>L_{0}$$ and is monotone increasing for $$0

This paper describes measurement methods of surface profiles that improve contact-type displacement sensor outputs by focusing on the contact point between the sphere tip of the sensor and the rough surface. We examined the geometry of a surface profile model and compared measurements using various methods with the measurement using a roughness meter. The spherical tip of the contact type displacement sensor touches the measurement surface and detects the displacement. The sphere tip radius of a typical contact-type displacement sensor ranges from 1–3 mm, causing the roughness curve to be “filtered” by the radius of the sphere. Three methods for estimating the valley portion of the surface profile are evaluated in this study: a) linear approximation of the concave portion of the surface profile, b) function approximation of the concave portion, and c) using the known nose radius of the machining tool. The following sphere tip radii were used to measure actual surface profiles: 0.25 mm, 0.5 mm, 1.0 mm and 1.5 mm. Given the conditions of the experimental model, we found that surface profiles with a roughness that approximates a predictable curve can be measured with a high degree of accuracy.

Let $M$ be a connected compact orientable surface, $f:M\to \mathbb{R}$ be a Morse function, and $h:M\to M$ be a diffeomorphism which preserves $f$ in the sense that $f\circ h = f$. We will show that if $h$ leaves invariant each regular component of each level set of $f$ and reverses its orientation, then $h^2$ is isotopic to the identity map of $M$ via $f$-preserving isotopy. This statement can be regarded as a foliated and a homotopy analogue of a well known observation that every reversing orientation orthogonal isomorphism of a plane has order $2$, i.e. a mirror symmetry with respect to some line. The obtained results hold in fact for a larger class of maps with isolated singularities from compact orientable surfaces to the real line and the circle.

The Brownian map is a random sphere-homeomorphic metric measure space obtained by "gluing together" the continuum trees described by the $x$ and $y$ coordinates of the Brownian snake. We present an alternative "breadth-first" construction of the Brownian map, which produces a surface from a certain decorated branching process. It is closely related to the peeling process, the hull process, and the Brownian cactus. Using these ideas, we prove that the Brownian map is the only random sphere-homeomorphic metric measure space with certain properties: namely, scale invariance and the conditional independence of the inside and outside of certain "slices" bounded by geodesics. We also formulate a characterization in terms of the so-called Levy net produced by a metric exploration from one measure-typical point to another. This characterization is part of a program for proving the equivalence of the Brownian map and Liouville quantum gravity with parameter $\gamma= \sqrt{8/3}$.

In this paper, we show that every homogeneous Finsler metric is a weakly stretch metric if and only if it reduces to a weakly Landsberg metric. This yields an extension of Tayebi–Najafi’s result that proved the result for the class of stretch Finsler metrics. Let $$F:=\alpha \phi (\beta /\alpha )$$ be a homogeneous weakly stretch $$(\alpha ,\beta )$$ -metric on a manifold M. We show that if $$\phi $$ is of polynomial type, then F is a Berwald metric. Also, we prove that F is a Berwald metric if and only if it has vanishing S-curvature. Then, we show that F is a Douglas metric if and only if it reduces to a Berwald metric. In continue, we show that every homogenous weakly stretch surface is a Landsberg surface. Finally, we characterize homogeneous weakly stretch spherically symmetric Finsler metrics.

A Riemann-Hilbert problem for a q-difference Painleve equation, known as $$q{\text {P}}_{{\text {IV}}}$$ , is shown to be solvable. This yields a bijective correspondence between the transcendental solutions of $$q{\text {P}}_{{\text {IV}}}$$ and corresponding data on an associated q-monodromy surface. We also construct the moduli space of $$q{\text {P}}_{{\text {IV}}}$$ explicitly.

We propose Hausdorff distance as a 3D aperture metric for the rough-walled 3D rock fracture. To verify its plausibility, we construct a fracture model from a 3D scanned crystalline rock sample. Comparison with the commonly used vertical aperture reveals significant advantages in imaging of the possible aperture bottlenecks present in the fracture volume. Another advantage stems from the omnidirectional nature of the Hausdorff distance as opposed to the directional true aperture estimation techniques developed previously. Furthermore, we compute aperture distributions for non-sheared and sheared fracture models, highlighting the differences and similarities between both aperture metrics. A parallel can be drawn with the surface roughness estimation method proposed by Grasselli, which also operates on the surface mesh constructed from the 3D scan point cloud. Hausdorff distance is shown to perform best in the highly sloped regions, where the full dip of the mesh polygon becomes significant and vertical aperture inevitably overestimates the true geometric aperture of the fracture.

Minkowski's classical existence theorem provides necessary and sufficient conditions for a Borel measure on the unit sphere of Euclidean space to be the surface area measure of a convex body. The solution is unique up to a translation. We deal with corresponding questions for unbounded convex sets, whose behavior at infinity is determined by a given closed convex cone. We provide an existence theorem and a stability result.

We derive the Faddeev-Reshetikhin (FR) model from a four-dimensional Chern- Simons theory with two order surface defects by following the work by Costello and Yamazaki [arXiv:1908.02289]. Then we present a trigonometric deformation of the FR model by employing a boundary condition with an R-operator of Drinfeld-Jimbo type. This is a generalization of the work by Delduc, Lacroix, Magro and Vicedo [arXiv:1909.13824] from the disorder surface defect case to the order one., 20 pages

Let X be a genus two double covering surface of the standard flat torus $$({\mathbb {C}}/{\mathbb {Z}}[i],dz)$$ with two ramified values $$0\ mod\ {\mathbb {Z}}[i]$$ and $$\lambda +i\mu \ mod\ {\mathbb {Z}}[i]$$ . We prove that if the equation $$l+m\lambda +n\mu =0$$ has nonzero integer solutions then the Hausdorff dimension of the set of nonergodic directions of X is either 0 or 1/2. And the precise criterion for the two possibilities is provided.

Given a finite subgroup G of the mapping class group of a surface S, the Nielsen realization problem asks whether G can be realized as a finite group of homeomorphisms of S. In 1983, Kerckhoff showed that for S a finite-type surface, any finite subgroup G may be realized as a group of isometries of some hyperbolic metric on S. We extend Kerckhoff's result to orientable, infinite-type surfaces. As applications, we classify torsion elements in the mapping class group of the plane minus a Cantor set, and also show that topological groups containing sequences of torsion elements limiting to the identity do not embed continuously into the mapping class group of S. Finally, we show that compact subgroups of the mapping class group of S are finite, and locally compact subgroups are discrete., Comment: v3 added results on (locally) compact subgroups of the mapping class group suggested by Mladen Bestvina. Also made minor edits according to the referee report. 8 pages

Mini-light-emitting diodes (mini-LEDs) are regarded as a promising light source for future high-end electronic products. Phosphors with a small size and high efficiency have been reported to achiev...