Non-compact Surfaces Research Articles

Noncompact surfaces, triangulations and rigidity

Abstract Every noncompact surface is shown to have a (3,6)‐tight triangulation, and applications are given to the generic rigidity of countable bar‐joint frameworks in . In particular, every noncompact surface has a (3,6)‐tight triangulation that is minimally 3‐rigid. A simplification of Richards' proof of Kerékjártó's classification of noncompact surfaces is also given.

Complete Solutions of Toda Equations and Cyclic Higgs Bundles Over Non-compact Surfaces

Abstract On a Riemann surface with a holomorphic $r$-differential, one can naturally define a Toda equation and a cyclic Higgs bundle with a grading. A solution of the Toda equation is equivalent to a harmonic metric of the Higgs bundle for which the grading is orthogonal. Here we focus on a general non-compact Riemann surface with an $r$-differential which is not necessarily meromorphic at infinity. We introduce the notion of complete solution of the Toda equation, and we prove the existence and uniqueness of a complete solution by using techniques for both Toda equations and harmonic bundles. Moreover, we show some quantitative estimates of the complete solution.

Renormalized volume, Polyakov anomaly, and orbifold Riemann surfaces

In [B. Taghavi Classical Liouville action and uniformization of orbifold Riemann surfaces, ], two of the authors studied the function 𝒮m=Sm−π∑i=1n(mi−1mi)loghi for orbifold Riemann surfaces of signature (g;m1,…,mne;np) on the generalized Schottky space Sg,n(m). In this paper, we prove the holographic duality between 𝒮m and the renormalized hyperbolic volume Vren of the corresponding Schottky 3-orbifolds with lines of conical singularity that reach the conformal boundary. In case of the classical Liouville action on Sg and Sg,n(∞), the holography principle was proved in [K. Krasnov, Holography and Riemann surfaces, , 929 (2000).], [J. Park , Potentials and Chern forms for Weil–Petersson and Takhtajan–Zograf metrics on moduli spaces, , 856 (2017).], respectively. Our result implies that Vren acts as a Kähler potential for a particular combination of the Weil-Petersson and Takhtajan-Zograf metrics that appears in the local index theorem for orbifold Riemann surfaces, as discussed by [L. A. Takhtajan and P. Zograf, Local index theorem for orbifold Riemann surfaces, .]. Moreover, we demonstrate that under the conformal transformations, the change of function 𝒮m is equivalent to the Polyakov anomaly, which indicates that the function 𝒮m is a consistent height function with a unique hyperbolic solution. Consequently, the associated renormalized hyperbolic volume Vren also admits a Polyakov anomaly formula. The method we used to establish this equivalence may provide an alternative approach to derive the renormalized Polyakov anomaly for Riemann surfaces with punctures (cusps), as described by [P. Albin , Ricci flow and the determinant of the Laplacian on non-compact surfaces, .]. Published by the American Physical Society 2025

A tale of bulk and branes: Symmetry TFT of 6D SCFTs from IIB/F-theory

We study the 7D Symmetry Topological Field Theory (SymTFT) associated to a 6D SCFT from the IIB/F-theory geometric engineering approach. The 6D (2,0) or (1,0) SCFT is constructed from IIB on a non-compact complex surface possibly with 7-branes. We derive the general form of 7D SymTFT actions from the compactification of IIB action on the boundary link of the base manifold of an elliptic Calabi-Yau threefold, for both the cases with or without flavor 7-branes intersecting the boundary link. Along the way we find new terms in the SymTFT action from the worldvolume action of flavor 7-branes involving the flavor center symmetries. We crosscheck the results against those obtained from either holographic constructions or the dual M-theory picture. Our construction potentially leads to a classification of the 7D SymTFTs which parallels the known geometric classification of the 6D SCFTs.

Counting homomorphisms from surface groups to finite groups

Abstract We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group G, where conjugacy classes of the boundary components of the surface must map to prescribed conjugacy classes in G, to a sum over values of irreducible characters of G weighted by Frobenius-Schur multipliers. The proof is structured so that the corresponding results for closed and possibly orientable surfaces, as well as some generalizations, are derived using the same methods. We then apply these results to the specific case of the symmetric group.

Leaf topology of minimal hyperbolic foliations with non simply-connected generic leaf

A noncompact (oriented) surface satisfies the condition ( ⋆ ) (\star ) if their isolated ends are accumulated by genus. We show that every surface satisfying this condition is homeomorphic to the leaf of a minimal codimension one foliation on a closed 3 3 -manifold whose generic leaf is not simply connected. Moreover, the above result is also true for any prescription of a countable family of noncompact surfaces (satisfying ( ⋆ ) (\star ) ): they can coexist in the same minimal codimension one foliation as above. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature.

Thurston’s compactification via geodesic currents: the case of non-compact finite area surfaces

In 1988, Bonahon gave a construction of Thurston's compactification of Teichmüller space using geodesic currents. His argument only applies in the case of closed surfaces, and there are good reasons for that. We present a variant which applies to surfaces of finite area and to do so we prove a control theorem for sequences of random geodesics. Note that this theorem may be of independant interest, especially when the surface is non-compact. Résumé. -En 1988, Bonahon a donné une preuve de la compactification de Thurston de l'espace de Teichmüller utilisant les courants géodésiques. Pour des raisons bien précises, cette preuve ne s'applique que dans le cas des surfaces fermées. On présente ici une variante des arguments de Bonahon qui s'applique aux surfaces d'aire finie grâce à un théorème de contrôle sur les suites de géodésiques aléatoires. Notons que ce théorème a son propre intérêt notamment lorsque la surface n'est pas compacte.

Estimates of Kähler metrics on noncompact finite volume hyperbolic Riemann surfaces, and their symmetric products

Estimates of Kähler metrics on noncompact finite volume hyperbolic Riemann surfaces, and their symmetric products

Simple infinite presentations for the mapping class group of a compact non-orientable surface

Omori and the author [R. Kobayashi and G. Omori, An infinite presentation for the mapping class group of a non-orientable surface with boundary, Osaka J. Math. 59(2) (2022) 269–314] have given an infinite presentation for the mapping class group of a compact non-orientable surface. In this paper, we give more simple infinite presentations for this group.

Every noncompact surface is a leaf of a minimal foliation

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed 3 -manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle. Moreover, the above result is also true for any prescription of a countable family of topologies of noncompact surfaces: they can coexist in the same minimal foliation. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature. Many oriented Seifert manifolds with a fibered incompressible torus and whose associated orbifold is hyperbolic admit minimal foliations as above. The given examples are not transversely C^{2} -smoothable.

Capillary surfaces: Stability, index and curvature estimates

Abstract In this paper, we investigate the connection between the index and the geometry and topology of capillary surfaces. We prove an index estimate for compact capillary surfaces immersed in general 3-manifolds with boundary. We also study noncompact capillary surfaces with finite index and show that, under suitable curvature assumptions, such surface is conformally equivalent to a compact Riemann surface with boundary, punctured at finitely many points. We then prove that a weakly stable capillary surface immersed in a half-space of R 3 \mathbb{R}^{3} which is minimal or has a contact angle less than or equal to π / 2 \pi/2 must be a half-plane. Using this uniqueness result, we obtain curvature estimates for strongly stable capillary surfaces immersed in a 3-manifold with bounded geometry.

E-series of character varieties of non-orientable surfaces

In this paper we are interested in two kinds of (stacky) character varieties associated to a compact non-orientable surface. (A) We consider the quotient stack of the space of representations of the fundamental group of this surface to GL(n). (B) We choose a set of k-punctures on the surface and a generic k-tuple of semisimple conjugacy classes of GL(n), and we consider the stack of anti-invariant local systems on the orientation cover of the surface with local monodromies around the punctures given by the prescribed conjugacy classes. We compute the number of points of these spaces over finite fields from which we get a formula for their E-series (a certain specialization of the mixed Poincare series). In case (B) we give a conjectural formula for the full mixed Poincare series.

Ginzburg-Landau equations on non-compact Riemann surfaces

We study the Ginzburg-Landau equations on line bundles over non-compact Riemann surfaces with constant negative curvature. We prove existence of solutions with energy strictly less than that of the constant curvature (magnetic field) one. These solutions are the non-commutative generalizations of the Abrikosov vortex lattice of superconductivity. Conjecturally, they are (local) minimizers of the Ginzburg-Landau energy. We obtain precise asymptotic expansions of these solutions and their energies in terms of the curvature of the underlying Riemann surface. Among other things, our result shows the spontaneous breaking of the gauge-translational symmetry of the Ginzburg-Landau equations.

МАССИВНЫЕ МНОЖЕСТВА, ПОРОЖДЁННЫЕ ПОЛУЛИНЕЙНЫМИ ЭЛЛИПТИЧЕСКИМИ ОПЕРАТОРАМИ НА НЕКОМПАКТНЫХ РИМАНОВЫХ МНОГООБРАЗИЯХ

One of the origins of the topic of this study is the classification theory of non-compact Riemannian surfaces. It is well known that on parabolic surfaces, any superharmonic functions bounded from below is the identical constant. Hyperbolic surfaces contain nontrivial superharmonic functions. This distinct property of parabolic surfaces form the basis for the definitions of parabolic manifolds with dimensions greater than two. The classification theory of Riemannian manifolds is directly related to Liouville-type theorems which assert the triviality of bounded solutions of elliptic equations. High efficiency in this topic was shown by the capacitive technique developed in the works of Grigoryan, Losev, Mazepa, and others. In particular, estimates were obtained for the dimensions of bounded harmonic functions and solutions of the stationary Schrödinger equation on noncompact Riemannian manifolds in terms of massive sets. In this paper, we study the properties of massive sets generated by a semilinear elliptic operator. It was possible to prove that the property of massiveness is preserved under variations of the potential. The current work generalizes or strengthens the results of Mazepa. A necessary condition for the existence of nontrivial bounded solutions of a semilinear equation is also obtained.

Geometrical compactifications of geodesic flows and path structures

In this paper, we construct a geometrical compactification of the geodesic flow of non-compact complete hyperbolic surfaces $$\Sigma $$ without cusps having finitely generated fundamental group. We study the dynamical properties of the compactified flow, for which we show the existence of attractive circles at infinity. The geometric structure of $${\text {T}}^1{\Sigma }$$ for which this compactification is realized is the pair of one-dimensional distributions tangent to the stable and unstable horocyles of $${\text {T}}^1{\Sigma }$$ . This is a Kleinian path structure, that is a quotient of an open subset of the flag space by a discrete subgroup $$\Gamma $$ of $$\text {PGL}_{3}\mathopen {(}\mathbb {R}\mathclose {)}$$ . Our study relies on a detailed description of the dynamics of $$\text {PGL}_{3}\mathopen {(}\mathbb {R}\mathclose {)}$$ on the flag space, and on the construction of an explicit fundamental domain for the action of $$\Gamma $$ on its maximal open subset of discontinuity in the flag space.

Shrinking target equidistribution of horocycles in cusps

Consider a shrinking neighborhood of a cusp of the unit tangent bundle of a noncompact hyperbolic surface of finite area, and let the neighborhood shrink into the cusp at a rate of T^{-1} as T rightarrow infty . We show that a closed horocycle whose length ell goes to infinity or even a segment of that horocycle becomes equidistributed on the shrinking neighborhood when normalized by the rate T^{-1} provided that T/ell rightarrow 0 and, for any delta >0, the segment remains larger than max left{ T^{-1/6},left( T/ell right) ^{1/2}right} left( T/ell right) ^{-delta }. We also have an effective result for a smaller range of rates of growth of T and ell . Finally, a number-theoretic identity involving the Euler totient function follows from our technique.

Complete Stationary Spacelike Surfaces in an n-Dimensional Generalized Robertson–Walker Spacetime

Several uniqueness results for non-compact complete stationary spacelike surfaces in an \(n(\ge 3)\)-dimensional Generalized Robertson Walker spacetime are obtained. In order to do that, we assume a natural inequality involving the Gauss curvature of the surface, the restrictions of the warping function and the sectional curvature of the fiber to the surface. This inequality gives the parabolicity of the surface. Using this property, a distinguished non-negative superharmonic function on the surface is shown to be constant, which implies that the stationary spacelike surface must be totally geodesic. Moreover, non-trivial examples of stationary spacelike surfaces in the four dimensional Lorentz–Minkowski spacetime are exposed to show that each of our assumptions is needed.

Non-positively curved Ricci Surfaces with catenoidal ends

A Ricci surface is defined to be a Riemannian surface \(({\varvec{M}},{\varvec{g}}_{\varvec{M}})\) whose Gauss curvature \({\varvec{K}}\) satisfies the differential equation \({\varvec{K}}\varvec{\Delta } {\varvec{K}} + {\varvec{g}}_{\varvec{M}}\left( {{\textbf {d}}{\varvec{K}}},{{\textbf {d}}{\varvec{K}}}\right) + {\textbf {4}}{\varvec{K}}^{\textbf {3}}={\textbf {0}}\). In the case where \({\varvec{K}}<{\textbf {0}}\), this equation is equivalent to the well-known Ricci condition for the existence of minimal immersions in \({\mathbb {R}}^3\). Recently, Andrei Moroianu and Sergiu Moroianu proved that a Ricci surface with non-positive Gauss curvature admits locally an isometric minimal immersion into \({\mathbb {R}}^3\). In this paper, we are interested in studying non-compact orientable Ricci surfaces with non-positive Gauss curvature. Firstly, we give a definition of catenoidal end for non-positively curved Ricci surfaces. Secondly, we develop a tool which can be regarded as an analogue of the Weierstrass data to obtain some classification results for non-positively curved Ricci surfaces of genus zero with catenoidal ends. Furthermore, we also give an existence result for non-positively curved Ricci surfaces of arbitrary positive genus which have finite catenoidal ends.

Chaotic behaviour of countable products of homeomorphism groups

ABSTRACT Relationships between a chaotic behaviour and closely related properties of topological transitivity, sensitivity to initial conditions, density of closed orbits of homeomorphism groups and their countable products are investigated. We provide a large number of new examples of chaotic groups of homeomorphisms of countable products of various metrizable topological spaces, including infinite-dimensional topological manifolds, whose factors can be as noncompact surfaces, so triangulable closed manifolds of an arbitrary dimension.

Geodesic families characterizing flat metrics on a cylinder and a plane

We prove that a complete non-compact surface contains a domain which is isometric to a pipe cylinder if all prime closed geodesics in it have the same length. As an application, we show that a flat cylinder is conjugacy rigid in the class of surfaces whose universal covering planes satisfy the divergence property. We study the divergence property from the view point of geodesic conjugacy for the Euclidean plane.