Proper Surfaces Research Articles

Algorithms for finding proper essential surfaces in 3-manifolds

In this paper, we present an algorithm which, for a given compact orientable irreducible boundary irreducible 3-manifold M, verifies whether M contains an essential orientable surface (possibly, with boundary), whose genus is at most N. The algorithm is based on Haken’s theory of normal surfaces, and on a trick suggested by Jaco and consisting in estimating the mean length of boundary curves in an unknown essential surface of a given genus in the given manifold.

Fotballkamp paa Grus og græs i Kristiania

Artiklen omhandler udviklingen i fodbold i Kristiania, med særlig fokus på fodboldbaner.
 
 Frode Fretland: Footballmatches on grass and gravel in Christiania
 This article will focus on the development of football and football grounds in the Norwegian capital Christiania until 1922. Questions about grounds for football have always been a central issue in the history of football in Norway. Due to difficult topography and the cold and wet climate in wintertime it has always been problematic to establish proper surfaces. In spite of bad facilities, association football has for a long time been the most popular sport, not in the capital alone, but all over the country. Football was from the very beginning seen as a way to offer children and young people a healthy and playful time of recreation in a natural environment. The municipal authorities of Christiania supported this by building grounds. Most of them were smaller gravel grounds (playing fields) in connection with schools, but only a few grounds were of good standard and had the right size for serious football matches. Really good football was played at full-sized grass pitches, but in Norway it was common until the end of 1910s to have gravel pitches only. When the national team played against Sweden and Denmark they always had difficulties playing well at the grass grounds because they were not used to it. At last, from 1918, a new grass ground near Christiania made it possible for the best local teams and the national team to train and play matches on a proper surface. This was followed by the first victories over Denmark and Sweden.

COMPLETE MINIMAL CYLINDERS PROPERLY IMMERSED IN THE UNIT BALL

The Calabi-Yau conjecture is one of the main problems in the global theory of complete minimal surfaces in R3. Francisco Martin and Santiago Morales have constructed complete proper minimal surfaces in convex bodies of R3. In this paper, we modify their technique in the cylindrical case, and construct a complete minimal cylinder properly immersed in the unit ball.

Complete proper minimal surfaces in convex bodies of $\mathbb R^3$, II. The behavior of the limit set

Let D be a regular, strictly convex bounded domain of \mathbb{R}^3 , and consider a Jordan curve \Gamma \subset \partial D . Then, for each \varepsilon>0 , we obtain the existence of a complete proper minimal immersion \psi_\varepsilon \colon \mathbb{D} \rightarrow D satisfying that the Hausdorff distance \delta^H(\psi_\varepsilon(\partial \mathbb{D}), \Gamma) < \varepsilon , where \psi_\varepsilon(\partial \mathbb{D}) represents the limit set of the minimal disk \psi_\varepsilon(\mathbb{D}) .

A three-dimensional inverse problem in estimating the internal heat flux of housing for high speed motors

The time-dependent heat flux generated in rotor and stator for the high speed electric motor is determined in this three-dimensional inverse heat conduction problem. The inverse algorithm utilizing the Steepest Descent Method (SDM) and a general purpose commercial code CFX4.4 is applied successfully in the present study in accordance with the simulated measured temperature distributions on some proper exterior surfaces. No cooling systems can be designed before the heat fluxes are estimated and identified. Two different functional forms for heat fluxes with different temperature measurement errors are used in the numerical experiments to illustrate the validity of the inverse algorithm. Results of the numerical simulation show that due to the structure of the cooling passages for motor housing, the estimated heat flux lying under the cooling passages is not accurate. However, when the concept of effective heat flux is applied, a reliable time-dependent heat flux can be obtained by using the present inverse algorithm.

Complete proper minimal surfaces in convex bodies of ℝ 3

Consider a convex domain B of ℝ 3 . We prove that there exist complete minimal surfaces that are properly immersed in B . We also demonstrate that if D and D ' are convex domains with D bounded and the closure of D contained in D ' , then any minimal disk whose boundary lies in the boundary of D can be approximated in any compact subdomain of D by a complete minimal disk that is proper in D ' . We apply these results to study the so-called type problem for a minimal surface: we demonstrate that the interior of any convex region of ℝ 3 is not a universal region for minimal surfaces, in the sense explained by Meeks and Pérez in [9].

SLANT IMMERSIONS OF COMPLEX SPACE FORMS AND CHEN'S INEQUALITY

A submanifold in a complex space form is called slant if it has constant Wirtinger angles. B. Y. Chen and Y. Tazawa proved that there do not exist minimal proper slant surfaces in CP2 and CH2. So it seems that the slant immersion has some interesting properties. The authors have great interest to consider slant immersions satisfying some additional conditions, such as unfull first normal bundles or Chen's equality holding. They prove that there do not exist n-dimensional Kaehlerian slant immersions in CPn and CHn with unfull first normal bundles. Next, it is seen that every Kaehlerian slant submanifold satisfying an equality of Chen is minimal which is similar to that of Lagrangian immersions. But in contrast, it is shown that a large class of slant immersions do not exist thoroughly. Finally, they give an application of Chen's inequality to general slant immersions in a complex projective space, which generalizes a result of Chen.

Electromagnetic waves over half-space metamaterials of arbitrary permittivity and permeability

Most electromagnetic problems deal with media with unit permeability. However, recent interest in metamaterials necessitated studies of wave characteristics in media with arbitrary permittivity and permeability whose real parts can be positive or negative. This paper presents analysis of wave characteristics on semiinfinite metamaterials. Waves are excited by electric or magnetic line sources, and the problem is separated into the p (TM) and the s (TE) polarization, showing symmetries. The Fourier spectra of the reflection and transmission coefficients are examined and the poles, branch points, and zeros are shown in the real /spl mu/-real /spl epsi/ diagram. We clarify the location of poles in proper and improper Riemann Surfaces, and the excitation of forward and backward surface waves, forward and backward Lateral waves, and Zenneck waves, and the relations between Brewster's angle and Sommerfeld poles. We include the behaviors of the backward surface waves and the temporal backward Lateral waves.

Proof of the oval conjecture for proper planar partition surfaces

We prove the ‘oval conjecture’ for planar partition functions, which says that the shift plane and the translation plane defined by a planar partition function form an oval pair of planes in the sense that each non-vertical line of one plane defines a topological oval in the projective closure of the other. The proof uses covering space techniques, and we have to assume that the generating function is proper in order to make those techniques available. As an application, we give a natural geometric construction of a homeomorphism between the Cartesian square of the shift line and its tangent bundle.

Existence of vector bundles and global resolutions for singular surfaces

We prove two results about vector bundles on singular algebraic surfaces. First, on proper surfaces there are vector bundles of rank two with arbitrarily large second Chern number and fixed determinant. Second, on sep- arated normal surfaces any coherent sheaf is the quotient of a vector bundle. As a consequence, for such surfaces the Quillen K-theory of vector bundles co- incides with the Waldhausen K-theory of perfect complexes. Examples show that, on nonseparated schemes, usually many coherent sheaves are not quo- tients of vector bundles.

On 3-dimensional contact slant submanifolds in Sasakian space forms

Recently, B.-Y. Chen obtained an inequality for slant surfaces in complex space forms. Further, B.-Y. Chen and one of the present authors proved the non-minimality of proper slant surfaces in non-flat complex space forms. In the present paper, we investigate 3-dimensional proper contact slant submanifolds in Sasakian space forms. A sharp inequality is obtained between the scalar curvature (intrinsic invariant) and the main extrinsic invariant, namely the squared mean curvature.It is also shown that a 3-dimensional contact slant submanifold M of a Sasakian space form M̆(c), with c ≠ 1, cannot be minimal.

New examples of reflectively symmetric minimal surfaces bounded by parallel straight lines

In this paper, we give examples of proper embedded Jenkins-Serrin type minimal surfaces with a reflective symmetry. That is, for each\( n \geq 2 \), we prove that there is a unique constant\( \phi _n, 0 \leq \phi _n < \pi \), and there exists a family of properly embedded minimal surfaces \( M(n, \phi), \phi_n < \phi < \pi\). Each \( M(n, \phi) \) is bounded by 2n parallel straight lines such that the interior of \( M(n, \phi) \) is the union of a minimal graph \( G(n,\phi) \) and its reflection. Each \( M(n,\phi) \)is invariant under \( D_{n}\times\mathbb{Z}_{2} \), where \( D_{n} \) is the general dihedral group, and\( \mathbb{Z}_{2} \) is generated by a reflection keeping each of the boundary lines invariant. The graph \( G(n,\phi) \) is over an non-convex bounded domain with a Jenkins-Serrin type capillary boundary values.¶¶Moreover, for each \( n\geq 2, M(n, \phi) \) can be put in a bigger family of immersed minimal surfaces, \( 0\leq \phi \leq \pi \). For \( 0 < \phi < \pi, M(n, \phi) \)has the same symmetric property and is bounded by parallel straight lines.¶¶When \( n\geq 2, G(n, \pi) \) is the Jenkins-Serrin graph over a domain bounded by a regular 2n-gon;\( M(2, 0) \) is a catenoid while for \( n\geq 3,M(n, 0) \) is the Jorge-Meeks n-noid; for\( 0 < \phi < \pi, M(2, \phi) \) is the KMR surface discovered by Karcher, and Meeks and Rosenberg.¶¶Thus in the moduli space of properly immersed minimal surfaces,\( M(n, \phi), n \geq 2, 0 \leq \phi \leq \pi \), is a connected path connecting the catenoid or Jorge-Meeks n-noid to the Jenkins-Serrin graph.

What is missing in canonical models for proper normal algebraic surfaces?

Smooth surfaces have finitely generated canonical rings and projective canonical models. For normal surfaces, however, the graded ring of multicanonical sections is possibly nonnoetherian, such that the corresponding homogeneous spectrum is noncompact. I construct a canonical compactification by adding finitely many non-ℚ-Gorenstein points at infinity, provided that each Weil divisor is numerically equivalent to a ℚ-Cartier divisor. Similar results hold for arbitrary Weil divisors instead of the canonical class.

There are enough Azumaya algebras on surfaces

Using Maruyama's theory of elementary transformations, I show that the Brauer group surjects onto the cohomological Brauer group for separated geometrically normal algebraic surfaces. As an application, I infer the existence of nonfree vector bundles on proper normal algebraic surfaces.

Normal del Pezzo surfaces containing a nonrational singularity

Working over perfect ground fields of arbitrary characteristic, I classify minimal normal del Pezzo surfaces containing a nonrational singularity. As an application, I determine the structure of 2-dimensional anticanonical models for proper normal algebraic surfaces. The anticanonical ring may be non-finitely generated. However, the anticanonical model is either a proper surface, or a proper surface minus a point.

Slant submanifolds of complex projective and complex hyperbolic spaces

An interesting class of submanifolds of Hermitian manifolds is the class of slant submanifolds which are submanifolds with constant Wirtinger angle. In (1-4,7,8) slant submanifolds of complex projective and complex hyperbolic spaces have been investigated. In particular, it was shown that there exist many proper slant surfaces in CP 2 and in CH 2 and many proper slant minimal surfaces in C 2 .I n contrast, in the first part of this paper we prove that there do not exist proper slant minimal surfaces in CP 2 and in CH 2 . In the second part, we present a general con- struction procedure for obtaining the explicit expressions of such slant submani- folds. By applying this general construction procedure, we determine the explicit expressions of special slant surfaces of CP 2 and of CH 2 . Consequently, we are able to completely determine the slant surface which satisfies a basic equality. Finally, we apply the construction procedure to prove that special -slant isometric immersions of a hyperbolic plane into a complex hyperbolic plane are not unique in general.

On contractible curves on normal surfaces

We give characterizations of contractible curves on proper normal algebraic surfaces in terms of complementary Weil divisors. From this we obtain some generalizations of the classical criteria for contractibility of Castelnuovo and Artin. Furthermore, we will derive a finiteness result on homogeneous spectra defined by Weil divisors on proper normal algebraic surfaces.

On non-projective normal surfaces

In this note we construct examples of non-projective normal proper algebraic surfaces and discuss the somewhat pathological behaviour of their Neron-Severi group. Our surfaces are birational to the product of a projective line and a curve of higher genus.

ON SLANT SURFACES

A slant immersion was introduced in [1] as an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. It is known that there exist ample examples of slant submanifolds; in particular, slant surfaces in complex-spaceforms. In this paper, we establish a sharp inequality for slant surfaces and determine the Riemannian structures of special slant surfaces in complexspace- forms. By applying the special forms of the Riemannian structures on special slant surfaces we prove that proper slant surfaces in C2 are minimal if and only if they are special slant. We also determine proper slant surfaces in complex-space-forms which satisfy the equality case of the inequality identically.

ON SLANT SURFACES

A slant immersion was introduced in [1] as an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. It is known that there exist ample examples of slant submanifolds; in particular, slant surfaces in complex-spaceforms. In this paper, we establish a sharp inequality for slant surfaces and determine the Riemannian structures of special slant surfaces in complexspace- forms. By applying the special forms of the Riemannian structures on special slant surfaces we prove that proper slant surfaces in C2 are minimal if and only if they are special slant. We also determine proper slant surfaces in complex-space-forms which satisfy the equality case of the inequality identically.