Cone Spline Surfaces and Spatial Arc Splines – a Sphere Geometric Approach

Basic sphere geometric principles are used to analyze approximation schemes of developable surfaces with cone spline surfaces, i.e., G ¹-surfaces composed of segments of right circular cones. These approximation schemes are geometrically equivalent to the approximation of spatial curves with G ¹-arc splines, where the arcs are circles in an isotropic metric. Methods for isotropic biarcs and isotropic osculating arc splines are presented that are similar to their Euclidean counterparts. Sphere geometric methods simplify the proof that two sufficiently close osculating cones of a developable surface can be smoothly joined by a right circular cone segment. This theorem is fundamental for the construction of osculating cone spline surfaces. Finally, the analogous theorem for Euclidean osculating circular arc splines is given.     

On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=p ⁿ are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath-Singerman symmetry. A classical theorem by Harnack states that the set of fixed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah-Singerman symmetry of Belyi surface with PSL(2,p) as a group of automorphisms has at most     

Some Remarks about the FM-partners of <Emphasis Type="Italic">K</Emphasis>3 Surfaces with Picard Numbers 1 and 2

In this paper we describe some results about K3 surfaces with Picard number 1 and 2. In particular, we give a new simple proof of a theorem due to Oguiso which shows that, given an integer N, there is a K3 surface with Picard number 2 and at least N non-isomorphic FM-partners. We describe also the Mukai vectors of the moduli spaces associated to the FM-partners of K3 surfaces with Picard number 1.     

Kawamata–Viehweg vanishing on rational surfaces in positive characteristic

We prove that the Kawamata–Viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to W ₂(k) of certain log pairs on smooth rational surfaces. As a corollary, the Kawamata–Viehweg vanishing theorem holds on log del Pezzo surfaces in positive characteristic.     

Integral Menger curvature for surfaces

We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a C1,λ-a priori bound for surfaces for which this functional is finite. In fact, it turns out that there is an explicit length scale R>0 which depends only on an upper bound E for the integral Menger curvature Mp(Σ) and the integrability exponent p, and not on the surface Σ itself; below that scale, each surface with energy smaller than E looks like a nearly flat disc with the amount of bending controlled by the (local) Mp-energy. Moreover, integral Menger curvature can be defined a priori for surfaces with self-intersections or branch points; we prove that a posteriori all such singularities are excluded for surfaces with finite integral Menger curvature. By means of slicing and iterative arguments we bootstrap the Hölder exponent λ up to the optimal one, λ=1−(8/p), thus establishing a new geometric ‘Morrey–Sobolev’ imbedding theorem.As two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral Menger curvature with a uniform bound on area, and of area minimizing surfaces subjected to a uniform bound on integral Menger curvature.     

Complete algebraic vector fields on affine surfaces,Part I

In this work we study the singularities at infinity of algebraic vector fields in dimension 2.These singularities will be classified under a mild assumption. The general problem is also reduced to the study of the combinatorics of certain resolutions which will be developed in Part II. Our main results are local and therefore can be carried over more general surfaces. Whereas we deal with ℂ -complete vector fields, the results also apply to ℝ-complete ones thanks to a theorem of Forstneric [7].     

Compact Kähler Surfaces with Trivial Canonical Bundle

The classical conjectures of Weil on K3 surfaces – that the set of suchsurfaces is connected; that a version of the Torelli theorem holds; thateach such surface is Kähler; and that the period map issurjective – are reconsidered in the light of a generalisation of theNakai–Moishezon criterion, and short proofs of all the conjectures aregiven. Most of the proofs apply equally or with minor variation tocomplex 2-tori, the only other compact Kähler surfaces with trivialcanonical bundle.     

Isoperimetric comparison theorems for manifolds with density

We give several isoperimetric comparison theorems for manifolds with density, including a generalization of a comparison theorem from Bray and Morgan. We find for example that in the Euclidean plane with radial density exp(r ^α ) for α ≥ 2, discs about the origin minimize perimeter for given area, by comparison with Riemannian surfaces of revolution.     

The closed embedded minimal surfaces in an almost positively curved three manifold

The compactness theorem of the closed embedded minimal surfaces of fixed genus in a 3-dimensional closed Riemannian manifoldN is proved, providedN is simply connected and the nonpositive value set of Ricci curvature is sufficiently concentrated within finite balls and the minimal surfaces are uniformly away from these balls.     

A generalization of the Pogorelov-Stocker theorem on complete developable surfaces

The well-known Pogorelov theorem stating the cylindricity of any C ¹-smooth, complete, developable surface of bounded exterior curvature in ℝ³ was generalized by Stocker to C ²-smooth surfaces with a more general notion of completeness. We extend Stocker’s result to C ¹-smooth surfaces that are normal developable in the Burago-Shefel’ sense. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 247–252, 2006.     

Dirac’s Theorem on Simplicial Matroids

We introduce the notion of k-hyperclique complexes, i.e., the largest simplicial complexes on the set [n] with a fixed k-skeleton. These simplicial complexes are a higher-dimensional analogue of clique (or flag) complexes (case k = 2) and they are a rich new class of simplicial complexes. We show that Dirac’s theorem on chordal graphs has a higher-dimensional analogue in which graphs and clique complexes get replaced, respectively, by simplicial matroids and k-hyperclique complexes. We prove also a higher-dimensional analogue of Stanley’s reformulation of Dirac’s theorem on chordal graphs.      

Curvature and uniformization

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e ^2u=K_o(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore ^2u giving the Poincaré metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem. Research supported in part by NSF Grant DMS-9971975 and also at MSRI by NSF grant DMS-9701755. Research supported in part by NSF Grant DMS-9877077     

On duality and fractionality of multicommodity flows in directed networks

In this paper we address a topological approach to multiflow (multicommodity flow) problems in directed networks. Given a terminal weight μ, we define a metrized polyhedral complex, called the directed tight span T_μ, and prove that the dual of the μ-weighted maximum multiflow problem reduces to a facility location problem on T_μ. Also, in case where the network is Eulerian, it further reduces to a facility location problem on the tropical polytope spanned by μ. By utilizing this duality, we establish the classifications of terminal weights admitting a combinatorial min–max relation (i) for every network and (ii) for every Eulerian network. Our result includes the Lomonosov–Frank theorem for directed free multiflows and Ibaraki–Karzanov–Nagamochi’s directed multiflow locking theorem as special cases.     

Dense Families of Selections and Finite-Dimensional Spaces

A characterization of n-dimensional spaces via continuous selections avoiding Z _n -sets is given, and a selection theorem for strongly countable-dimensional spaces is established. We apply these results to prove a generalized Ostrand's theorem, and to obtain a new alternative proof of the Hurewicz formula. It is also shown that our selection theorem yields an easy proof of a Michael's result.     

Infinitesimal Torelli theorem for surfaces of general type with certain invariants

We prove the infinitesimal Torelli theorem for general minimal complex surfaces X's with the first Chern number 3, geometric genus 1, and irregularity 0 which have non-trivial 3-torsion divisors. We also show that the coarse moduli space for surfaces with the invariants as above is a 14-dimensional unirational variety.     

Exotic Minimal Surfaces

We prove a general fusion theorem for complete orientable minimal surfaces in ?³ with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are produced. More specifically, universal surfaces (i.e., surfaces from which all minimal surfaces can be recovered) and space-filling surfaces with arbitrary genus and no symmetries.     

A Characterization of Counterexamples to the Kodaira-Ramanujam Vanishing Theorem on Surfaces in Positive Characteristic

The author gives a characterization of counterexamples to the Kodaira-Ramanujam vanishing theorem on smooth projective surfaces in positive characteristic. More precisely, it is reproved that if there is a counterexample to the Kodaira-Ramanujam vanishing theorem on a smooth projective surface X in positive characteristic, then X is either a quasi-elliptic surface of Kodaira dimension 1 or a surface of general type. Furthermore, it is proved that up to blow-ups, X admits a fibration to a smooth projective curve, such that each fiber is a singular curve.     

A Zoll Counterexample to a Geodesic Length Conjecture

We construct a counterexample to a conjectured inequality L ≤ 2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin’s theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd deformation of the round metric. Thus the round metric is not optimal for the ratio L/D. F.B. supported by the Swiss National Science Foundation. C.C. supported by NSF grants DMS 02-02536 and DMS 07-04145. M.K. supported by the Israel Science Foundation (grants 84/03 and 1294/06)     

A Criterion for Designs in ℤ4-codes on the Symmetrized Weight Enumerator

The Assmus–Mattson theorem is known as a method to find designs in linear codes over a finite field. It is an interesting problem to find an analog of the theorem for Z ₄-codes. In a previous paper, the author gave a candidate of the theorem. The purpose of this paper is to give an improvement of the theorem. It is known that the lifted Golay code over Z ₄ contains 5-designs on Lee compositions. The improved method can find some of those without computational difficulty and without the help of a computer.     

On two-dimensional parametric variational problems

Let be a two-dimensional parametric variational integral the Lagrangian F(x,z) of which is positive definite and elliptic, and suppose that is a closed rectifiable Jordan curve in . We then prove that there is a conformally parametrized minimizer of in the class of surfaces of the type of the disk B which are bounded by . An immediate consequence of this theorem is that the Dirichlet integral and the area functional have the same infima, a result whose proof usually requires a Lichtenstein-type mapping theorem or else Morrey's lemma on -conformal mappings. In addition we show that the minimizer of is H?lder continuous in B, and even in if satisfies a chord-arc condition. In Section 1 it is described how our results are related to classical investigations, in particular to the work of Morrey. Without difficulty our approach can be carried over to two-dimensional surfaces of codimension greater than one. Received July 20, 1998 / Accepted October 23, 1998