Minimal surfaces in R3 with one end¶and bounded curvature

We study complete minimal surfaces M immersed in R ³, with finite topology and one end. We give conditions which oblige M to be conformally a compact Riemann surface punctured in one point, and we show that M can be parametrized by meromorphic data on this compact Riemann surface. The goal is to prove that when M is also embedded, then the end of M is asymptotic to an end of a helicoid (or M is a plane). Received: 13 January 1997 / Revised version: 15 September 1997     

Enneper immersions

We describe a method to construct minimal surfaces inR ³ which has many computational simplicities and prove that any immersed minimal surface inR ³ may be constructed by using that method. We also show that under certain finite hypotheses on the orders of the coordinate functions, among all conformal minimal immersions of the plane with Gauss mapg=e ^az+b only the parametrizations of helicoids and planes are proper embeddings.     

The Chen-Type of the Spiral Surfaces

We show that a spiral surface M in E³ is of finite type if and only if M is minimal Also, the plane is the only spiral surface in E³ whose the Gauss map G is of finite type, or satisfies the condition ΔG = ΛG, where Λ ∈ R^3×3.     

Minimal Surfaces Obtained by Ribaucour Transformations

We consider Ribaucour transformations between minimal surfaces and we relate such transformations to generating planar embedded ends. Applying Ribaucour transformations to Enneper's surface and to the catenoid, we obtain new families of complete, minimal surfaces, of genus zero, immersed in R ³, with infinitely many embedded planar ends or with any finite number of such ends. Moreover, each surface has one or two nonplanar ends. A particular family is obtained from the catenoid, for each pair (n,m), nm, such that n m0 is an irreducible rational number. For any such pair, we get a 1-parameter family of finite total curvature, complete minimal surfaces with n+2 ends, n embedded planar ends and two nonplanar ends of geometric index m, whose total curvature is –4(n+m). The analytic interpretation of a Ribaucour transformation as a Bäcklund type transformation and a superposition formula for the nonlinear differential equation = e^-2 is included.     

Complete harmonic stable minimal hypersurfaces in a Riemannian manifold

In this paper, we will introduce the notion of harmonic stability for complete minimal hypersurfaces in a complete Riemannian manifold. The first result we prove, is that a complete harmonic stable minimal surface in a Riemannian manifold with non-negative Ricci curvature is conformally equivalent to either a plane R ² or a cylinder R × S ¹, which generalizes a theorem due to Fischer-Colbrie and Schoen [12]. The second one is that an n ≥ 2-dimensional, complete harmonic stable minimal, hypersurface M in a complete Riemannian manifold with non-negative sectional curvature has only one end if M is non-parabolic. The third one, which we prove, is that there exist no non-trivial L ²-harmonic one forms on a complete harmonic stable minimal hypersurface in a complete Riemannian manifold with non-negative sectional curvature. Since the harmonic stability is weaker than stability, we obtain a generalization of a theorem due to Miyaoka [20] and Palmer [21]. Research partially Supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The author’s research was supported by grant Proj. No. KRF-2007-313-C00058 from Korea Research Foundation, Korea. Authors’ addresses: Qing-Ming Cheng, Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan; Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea     

Covering a plane with ellipses

Abstract

This paper is devoted to the construction of regular min-density plane coverings with ellipses of one, two and three types. This problem is relevant, for example, to power-efficient surface sensing by autonomous above-grade sensors. A similar problem, for which discs are used to cover a planar region, has been well studied. On the one hand, the use of ellipses generalizes a mathematical problem; on the other hand, it is necessary to solve these types of problems in real applications of wireless sensor networks. This paper both extends some previous results and offers new regular covers that use a small number of ellipses to cover each regular polygon; these covers are characterized by having minimal known density in their classes and give the new upper bounds for densities in these classes as well.     

Surfaces à points doubles isolés

We give a characterization of the generic projection on P ² of an algebraic surface of P ³ with a finite number of nodes. The construction of an algebraic surface of P ³ with a given number of nodes is thus equivalent to the construction of a plane curve with nodes and cusps in some special position. Received: November 9, 1996     

Undesirable facility location with minimal covering objectives

An undesirable facility is to be located within some feasible region of any shape in the plane or on a planar network. Population is supposed to be concentrated at a finite number n of points. Two criteria are taken into account: a radius of influence to be maximised, indicating within which distance from the facility population disturbance is taken into consideration, and the total covered population, i.e. lying within the influence radius from the facility, which should be minimised. Low complexity polynomial algorithms are derived to determine all nondominated solutions, of which there are only O(n³) for a fixed feasible region or O(n²) when locating on a planar network. Once obtained, this information allows almost instant answers and a trade-off sensitivity analysis to questions such as minimising the population within a given radius (minimal covering problem) or finding the largest circle not covering more than a given total population.     

An embedding theorem for finite planar spaces

LetS be a finite planar space such that any two distinct planes intersect in a line. We show thatk≤n ²+1 for anyk-cap ofS, wheren is the order ofS. Moreover, if a (n ²+1)-cap exists inS, a necessary and sufficient condition is provided forS to be embeddable in a 3-dimensional projective space. Work supported by the National Research Project “Strutture geometriche, Combinatoria e loro applicazioni” of the italian M.U.R.S.T.     

Some remarks on complete simply connected minimal surfaces meeting the planes x3 = Constant Transversally

The helicoid and plane are the only known complete embedded minimal surfaces inR ³ that are simply connected. We prove the helicoid and plane are the only surfaces of this type that have bounded curvature and meet each plane x₃ = constant in (at most) one smooth connected curve.     

Commutative presemifields and semifields

Strong conditions are derived for when two commutative presemifields are isotopic. It is then shown that any commutative presemifield of odd order can be described by a planar Dembowski-Ostrom polynomial and conversely, any planar Dembowski-Ostrom polynomial describes a commutative presemifield of odd order. These results allow a classification of all planar functions which describe presemifields isotopic to a finite field and of all planar functions which describe presemifields isotopic to Albert's commutative twisted fields. A classification of all planar Dembowski-Ostrom polynomials over any finite field of order p³, p an odd prime, is therefore obtained. The general theory developed in the article is then used to show the class of planar polynomials X¹⁰+aX⁶−a²X² with a≠0 describes precisely two new commutative presemifields of order e³ for each odd e?5.     

The geometry of minimal surfaces of finite genus II; nonexistence of one limit end examples

We demonstrate that a properly embedded minimal surface in ³ with finite genus cannot have one limit end. Mathematics Subject Classification (1991) 53A10, 49Q05, 53C42     

A Note on Shiffman's Theorems

Shiffman proved his famous first theorem, that if A R³ is a compact minimal annulus bounded by two convex Jordan curves in parallel (say horizontal) planes, then A is foliated by strictly convex horizontal Jordan curves. In this article we use Perron's method to construct minimal annuli which have a planar end and are bounded by two convex Jordan curves in horizontal planes, but the horizontal level sets of the surfaces are not all convex Jordan curves or straight lines. These surfaces show that unlike his second and third theorems, Shiffman's first theorem is not generalizable without further qualification.     

Two graphs without planar covers

In this note we prove that two specific graphs do not have finite planar covers. The graphs are K₇–C₄ and K_4,5–4K₂. This research is related to Negami's 1‐2‐∞ Conjecture which states “A graph G has a finite planar cover if and only if it embeds in the projective plane.” In particular, Negami's Conjecture reduces to showing that 103 specific graphs do not have finite planar covers. Previous (and subsequent) work has reduced these 103 to a few specific graphs. This paper covers 2 of the remaining cases. The sole case currently remaining is to show that K_2,2,2,1 has no finite planar cover. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 318–326, 2002     

On a characteristic property of minimal-ruled surfaces

LetS be a surface of classC ⁴ in 3-dimensional Euclidean space. In this paper it is shown that any two of the following three conditions imply the third one: (a)S is a minimal surface, (b) Two families of Laguerre lines ofS form a conjugate net, (c)S is a non-developable ruled surface. Furthermore, it is proved that any surface (other than a sphere, a cylinder of revolution and a plane) of constant mean curvature on which the two families of Laguerre lines form a conjugate net is a minimal-helicoid.     

Planar functions over fields of characteristic two

Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over \(\mathbb {Z}_{4}\) . We then specialise to planar monomial functions f(x)=cx ^t and present constructions and partial results towards their classification. In particular, we show that t=1 is the only odd exponent for which f(x)=cx ^t is planar (for some nonzero c) over infinitely many fields. The proof techniques involve methods from algebraic geometry.     

Some remarks on dualisability and endodualisability

The fundamental problem of dualisability and the particular problem of endodualisability are discussed. It is proved tha every finite generating algebra of a quasi-variety generated by a finite dualisable algebra D is also dualisable. The corresponding result for endodualisability is true when D is subdirectly irreducible. Under special conditions, it is also proved that a finite algebra M is endodualisable if and only if any finite power M ⁿ of M is endodualisable. Received January 27, 1999; accepted in final form September 17, 1999.     

Total Scalar Curvature and L 2 Harmonic 1-Forms on a Minimal Hypersurface in Euclidean Space

Let M ⁿ , n 3, be a complete oriented immersed minimal hypersurface in Euclidean space R ⁿ⁺¹. We show that if the total scalar curvature on M is less than the n/2 power of 1/C _s , where C _s is the Sobolev constant for M, then there are no L ² harmonic 1-forms on M. As corollaries, such a minimal hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This implies finally that M is a hyperplane.     

Isometric embeddings of locally Euclidean metrics in ℝ3 as conical surfaces

It is proved that if a domain with a locally Euclidean metric can be isometrically immersed in the Euclidean plane ?² with the standard metric, then it can be isometrically embedded in ?³ as a conical surface whose projection on a sphere centered at the vertex of the cone is a self-avoiding planar graph with sufficiently smooth edges of specially selected lengths.     

Optimal boundary control of the wave equation with pointwise control constraints

In optimal control problems frequently pointwise control constraints appear. We consider a finite string that is fixed at one end and controlled via Dirichlet conditions at the other end with a given upper bound M for the L ^∞-norm of the control. The problem is to control the string to the zero state in a given finite time. If M is too small, no feasible control exists. If M is large enough, the optimal control problem to find an admissible control with minimal L ²-norm has a solution that we present in this paper.