Spline Smoothing on Surfaces

This article presents a method for estimating functions on topologically and/or geometrically complex surfaces from possibly noisy observations. Our approach is an extension of spline smoothing, using a finite element method. The article has a substantial tutorial component: we start by reviewing smoothness measures for functions defined on surfaces, simplicial surfaces and differentiable structures on such surfaces, subdivison functions, and subdivision surfaces. After describing our method, we show results of an experiment comparing finite element approximations to exact smoothing splines on the sphere, and we give examples suggesting that generalized cross-validation is an effective way of determining the optimal degree of smoothing for function estimation on surfaces.     

Divergence-Free RBFs on Surfaces

This article presents a new tool for fitting a divergence-free vector field tangent to a two-dimensional orientable surface to samples of such a field taken at scattered sites on P. This method, which involves a kernel constructed from radial basis functions, has applications to problems in geophysics, and has the advantage of avoiding problems with poles. Numerical examples testing the method on the sphere are included.     

Noether's Theorem on Surfaces

In this article we prove a version of Noether's Theorem (of Calculus of Variations) which is valid for a general regular (compact) surface. As a special feature, the Lie group of transformations is allowed to act on the Cartesian product of the surface and the functional space. Additionally, we apply the Theorem to a problem in Classical Differential Geometry of surfaces. The given application is actually an example showing how Noether's Theorem can be used to construct invariant properties of the solutions to variational problems defined on surfaces, or equivalently, of the solutions to the associated Euler-Lagrange equations resulting from them.     

极小曲面中的若干问题

本文综述了Ｅ＾ｎ和Ｓ＾ｎ中极小曲面的若干经典结果和最新发展，指出了一些尚未解决的问题，在第４节中，对Ｅ＾３中极小曲面的Ｆｕｊｉｍｏｔｏ定理给出了一个更直接的证明。     

Cycles on Arithmetic Surfaces

We develop a localized intersection theory for arithmetic schemes on the model of Fulton's intersection theory. We prove a Lefschetz fixed point formula for arithmetic surfaces, and give an application to a conjecture of Serre on the existence of Artin's representations for regular local rings of dimension 2 and unequal characteristic.     

Equimatchable Graphs on Surfaces

A graph G is equimatchable if each matching in G is a subset of a maximum‐size matching and it is factor critical if has a perfect matching for each vertex v of G. It is known that any 2‐connected equimatchable graph is either bipartite or factor critical. We prove that for 2‐connected factor‐critical equimatchable graph G the graph is either or for some n for any vertex v of G and any minimal matching M such that is a component of . We use this result to improve the upper bounds on the maximum number of vertices of 2‐connected equimatchable factor‐critical graphs embeddable in the orientable surface of genus g to if and to if . Moreover, for any nonnegative integer g we construct a 2‐connected equimatchable factor‐critical graph with genus g and more than vertices, which establishes that the maximum size of such graphs is . Similar bounds are obtained also for nonorientable surfaces. In the bipartite case for any nonnegative integers g, h, and k we provide a construction of arbitrarily large 2‐connected equimatchable bipartite graphs with orientable genus g, respectively nonorientable genus h, and a genus embedding with face‐width k. Finally, we prove that any d‐degenerate 2‐connected equimatchable factor‐critical graph has at most vertices, where a graph is d‐degenerate if every its induced subgraph contains a vertex of degree at most d.     

Bonnet Surfaces and Isothermic Surfaces

By the study of the relations between Bonnet surfaces and isothermic surfaces, we obtain classification results of Bonnet surfaces in 3-dimensional space form R³(c) and of the spacelike Bonnet surfaces in indefinite space form R₁ ³(c), which generalize the results in Bobenko’s [1] and Peng-Lu’s [11]. It is remarkable that there exist always Bonnet surfaces which are not Weingarten surfaces, if the ambient space is not R³(c) for c ≥ 0.     

Simple Curves on Surfaces

We study simple closed geodesics on a hyperbolic surface of genus g with b geodesic boundary components and c cusps. We show that the number of such geodesics of length at most L is of order L^6g+2b+2c–6. This answers a long-standing open question.     

Testing Graph Isotopy on Surfaces

We investigate the following problem: Given two embeddings G ₁ and G ₂ of the same abstract graph G on an orientable surface S, decide whether G ₁ and G ₂ are isotopic; in other words, whether there exists a continuous family of embeddings between G ₁ and G ₂. We provide efficient algorithms to solve this problem in two models. In the first model, the input consists of the arrangement of G ₁ (resp., G ₂) with a fixed graph cellularly embedded on S; our algorithm is linear in the input complexity, and thus, optimal. In the second model, G ₁ and G ₂ are piecewise-linear embeddings in the plane, minus a finite set of points; our algorithm runs in O(n ^3/2logn) time, where n is the complexity of the input. The graph isotopy problem is a natural variation of the homotopy problem for closed curves on surfaces and on the punctured plane, for which algorithms have been given by various authors; we use some of these algorithms as a subroutine. As a by-product, we reprove the following mathematical characterization, first observed by Ladegaillerie (Topology 23:303–311, 1984): Two graph embeddings are isotopic if and only if they are homotopic and congruent by an oriented homeomorphism.     

Modelling 3-configurations on Surfaces

A 3-configuration is a finite geometry satisfying axioms: (i) each line contains exactly 3 points; (ii) each point is on exactly r lines, where r is a fixed positive integer; and (iii) each pair of distinct points are on at most one common line. Such geometries correspond to K_3⁻ decompositions of their Menger graphs, and hence to bichromatic dual surface (or pseudosurface) imbeddings of these graphs where one color class consists of triangular regions modelling the lines of the geometry. Such imbeddings have been found for all 320 pairs (v, r), where v is the number of points in the geometry, satisfying 2r + 1 ≤ v ≤ 50 and vr ≡ 0 (mod 3). Here we discuss some of the more interesting among these.     

Harmonic Maps on Hadamard Surfaces

Suppose (,g) is a simply connected, complete and noncompact surface. We call it is aHadamard surface if its Gaussian curwature is between two negative constants. By the Uniformization Theorem, we know that such surfaces are always conformal to 2-dAn hyperbolicspaces (D, ds2p). Here D is the unit disc on the plane or the upper half plane, and ds2p is thePoincare metric on it. In this note, we are interested in the Dirichlet problem of harmonic mapson such surfaces.Due to the conformal invar…     

Circle Patterns on Singular Surfaces

We consider “hyperideal” circle patterns, i.e., patterns of disks appearing in the definition of the weighted Delaunay decomposition associated with a set of disjoint disks, possibly with cone singularities at the centers of those disks. Hyperideal circle patterns are associated with hyperideal hyperbolic polyhedra. We describe the possible intersection angles and singular curvatures of those circle patterns on Euclidean or hyperbolic surfaces with cone singularities. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra. The results presented here extend those in Schlenker (Math. Res. Lett. 12(1), 82–112, [2005]), however, the proof is completely different (and more intricate) since Schlenker (Math. Res. Lett. 12(1), 82–112, [2005]) used a shortcut which is not available here. The author would like to thank the RIP program at Oberwolfach, where part of the research presented here was conducted. Partially supported by the “ACI Jeunes Chercheurs” Métriques privilégiés sur les variétés à bord, 2003-06, and the ANR program Representations of surface groups, 2007-09.     

Boxicity of Graphs on Surfaces

The boxicity of a graph G = (V, E) is the least integer k for which there exist k interval graphs G _i  = (V, E _i ), 1 ≤ i ≤ k, such that ${E = E_1 \cap \cdots \cap E_k}$ . Scheinerman proved in 1984 that outerplanar graphs have boxicity at most two and Thomassen proved in 1986 that planar graphs have boxicity at most three. In this note we prove that the boxicity of toroidal graphs is at most 7, and that the boxicity of graphs embeddable in a surface Σ of genus g is at most 5g + 3. This result yields improved bounds on the dimension of the adjacency poset of graphs on surfaces.     

Bihermitian Structures on Complex Surfaces

Bihermitian complex surfaces are oriented conformal four-manifoldsadmitting two independent compatible complex structures. Non-anti-self-dualbihermitian structures on R⁴ and the four-dimensional torusT⁴ have recently been discovered by P. Kobak. We show that anoriented compact 4-manifold, admitting a non-anti-self-dualbihermitian structure, is a torus or K3 surface in the stronglybihermitian case (when the two complex structures are independentat each point) or, otherwise, must be obtained from the complexprojective plane or a minimal ruled surface of genus less than2 by blowing up points along some anti-canonical divisor (butthe actual existence of bihermitian structures in the lattercase is still an open question). The paper includes a generalmethod for constructing non-anti-self-dual bihermitian structureson tori, K3 surfaces and S¹ x S³. Further properties of compactbihermitian surfaces are also investigated. 1991 MathematicsSubject Classification: 53C12, 53C55, 32J15.     

Discrete Welding on Riemann Surfaces

The process of conformal welding had been investigated for nearly 50 years. In recent years welding has drawn increased interest with applications ranging from image recognition to string theory. We describe a discrete form of conformal welding for Riemann surfaces using circle packings and show how it can be used to approximate a continuous welding.     

Finite Index Operators on Surfaces

We consider differential operators L acting on functions on a Riemannian surface, Σ, of the form $$L = \Delta+ V -a K,$$ where Δ is the Laplacian of Σ, K is the Gaussian curvature, a is a positive constant, and V∈C ^∞(Σ). Such operators L arise as the stability operator of Σ immersed in a Riemannian three-manifold with constant mean curvature (for particular choices of V and a). We assume L is nonpositive acting on functions compactly supported on Σ. If the potential, V:=c+P with c a nonnegative constant, verifies either an integrability condition, i.e., P∈L ¹(Σ) and P is nonpositive, or a decay condition with respect to a point p ₀∈Σ, i.e., |P(q)|≤M/d(p ₀,q) (where d is the distance function in Σ), we control the topology and conformal type of Σ. Moreover, we establish a Distance Lemma. We apply such results to complete oriented stable H-surfaces immersed in a Killing submersion. In particular, for stable H-surfaces in a simply-connected homogeneous space with 4-dimensional isometry group, we obtain:

• There are no complete stable H-surfaces Σ??²×?, H>1/2, so that either $K_{e}^{+}:=\max \left \{0,K_{e}\right \} \in L^{1} (\Sigma)$ or there exist a point p ₀∈Σ and a constant M so that |K _e (q)|≤M/d(p ₀,q); here K _e denotes the extrinsic curvature of Σ.
• Let $\Sigma\subset \mathbb{E}(\kappa, \tau)$ , τ≠0, be an oriented complete stable H-surface so that either ν ²∈L ¹(Σ) and 4H ²+κ≥0, or there exist a point p ₀∈Σ and a constant M so that |ν(q)|²≤M/d(p ₀,q) and 4H ²+κ>0. Then:
• In $\mathbb{S}^{3}_{\text{Berger}}$ , there are no such a stable H-surfaces.
• In Nil₃, H=0 and Σ is either a vertical plane (i.e., a vertical cylinder over a straight line in ?²) or an entire vertical graph.
• In $\widetilde{\mathrm{PSL}(2,\mathbb{R})}$ , $H=\sqrt{-\kappa }/2$ and Σ is either a vertical horocylinder (i.e., a vertical cylinder over a horocycle in ?²(κ)) or an entire graph.

     

Vortex Filament Method

In this paper, the convergence of the vortex filament methodfor three-dimensional incompressible and inviscid fluid flowis proved. Properties of consistency and stability are analysed.The foundation for studying these properties is based on thecubature formulae with lines as well as on the specific useof the vector measure that transports the vorticity by the flow,preserving the filament structure of the solution of the problem.In this way, the method takes into account the stretching termimplicitly.     

Vortex Perturbation Dynamics

An initial value approach is used to examine the dynamics of perturbations introduced into a vortex under strain. Both the basic vortex considered and the perturbations are taken as three-dimensional. An explicit solution for the time evolution of the vorticity perturbations is given for arbitrary initial vorticity. Analytical solutions for the resulting velocity components are found when the initial vorticity is assumed to be localized. For more general initial vorticity distributions, the velocity components are determined numerically. It is found that the variation in the radial direction of the initial vorticity disturbance is the most important factor influencing the qualitative behavior of the solutions. Transient growth in the magnitude of the velocity components is found to be directly attributable to the compactness of the initial vorticity.     

Integrable Geodesic Flows on Surfaces

We propose a new condition à{{\aleph}} which enables us to get new results on integrable geodesic flows on closed surfaces. This paper has two parts. In the first, we strengthen Kozlov’s theorem on non-integrability on surfaces of higher genus. In the second, we study integrable geodesic flows on a 2-torus. Our main result for a 2-torus describes the phase portraits of integrable flows. We prove that they are essentially standard outside what we call separatrix chains. The complement to the union of the separatrix chains is C ⁰-foliated by invariant sections of the bundle.