On Minimal Annuli with a Straight Line Boundary

Shiffman proved that if a minimal annulus A in a slab is bounded by two convex Jordan curves contained respectively in the two boundary planes P and Q of the slab, then A intersects all parallel planes between P and Q in strictly convex curves. We generalize Shiffman's result to the case that A is bounded by a strictly convex C² Jordan curve and a straight line. We show that in this case Shiffman's result is still true.     

The uniqueness for minimal surfaces inS 3

We give a sufficient condition on a Jordan curve in the 3-dimensional open hemisphereH ofS ³ in terms of the Hopf fibering under which spans a unique compact generalized minimal surface inH. The maximum principle for minimal surfaces inS ³ is proved and plays an important role in the proof of the uniqueness theorem.Dedicated to Professor Shingo Murakami on his 60th birthdayThis work was carried out while the author was a visitor to the Max-Planck-Institut für Mathematik.     

Some existence results about radial graphs of constant mean curvature with boundary in parallel planes

In this work, for given H, we investigate the existence of radial cmc H annulus spanning two given non necessarily convex Jordan curves in parallel planes of . We established some existence results under hypotheses relating the geometry of the curves and the distance between the planes. Ari J. Aiolfi was partially supported by Fapergs and Programa FIPE Junior/UFSM.     

Existence and Uniqueness of F-Minimal Surfaces

We extend a classical result of Radó and Kneser concerning uniqueness ofminimal surfaces bounded by a given closed Jordan curve in³ to the case of extremals for certain geometric variationalintegrals. Using standard elliptic PDE theory, this gives the existenceand uniqueness of embedded F-minimal surfaces for suitable boundarycurves that project simply onto the boundary of a plane convex domain.     

Mean convex hulls and least area disks spanning extreme curves

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in such that for any asymptotic curve , there is a canonical mean convex hull containing all minimal planes spanning Γ. Applying this to quasi-Fuchsian manifolds, we show that for any quasi-Fuchsian manifold, there exist a canonical mean convex core capturing all essential minimal surfaces. On the other hand, we also show that for a generic C³-smooth curve in the boundary of C³-smooth mean convex domain in ℝ³, there exist a unique least area disk spanning the curve.     

An asymptotic theorem for minimal surfaces and existence results for minimal graphs in $${\mathbb H^2 \times \mathbb R}$$

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in . As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary Γ_∞ is a Jordan curve homologous to zero in such that Γ_∞ is contained in a slab between two horizontal circles of with width equal to π. We construct vertical minimal graphs in over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains Ω in are non necessarily convex and non necessarily bounded by convex arcs; each component of its boundary is properly embedded with zero, one or two points on its asymptotic boundary, satisfying a further geometric condition. The first author wish to thank Laboratoire Géométrie et Dynamique de l’Institut de Mathématiques de Jussieu for the kind hospitality and support. The authors would like to thank CNPq, PRONEX of Brazil and Accord Brasil-France, for partial financial support.     

Pseudo Jordan domains and reflecting Brownian motions

Summary The manifold metric between two points in a planar domain is the minimum of the lengths of piecewiseC ¹ curves in the domain connecting these two points. We define a bounded simply connected planar region to be a pseudo Jordan domain if its boundary under the manifold metric is topologically homeomorphic to the unit circle. It is shown that reflecting Brownian motionX on a pseudo Jordan domain can be constructed starting at all points except those in a boundary subset of capacity zero.X has the expected Skorokhod decomposition under a condition which is satisfied when G has finite 1-dimensional lower Minkowski content.     

Shortest paths on convex hypersurfaces of Riemannian spaces

A convex hypersurface in a Riemannian space M^m is part of the boundary of an m-dimensional locally convex set. It is established that there exists an intrinsic metric of such a hypersurface and it has curvature which is bounded below in the sense of A. D. Aleksandrov; curves with bounded variation of rotation in are shortest paths in M^m. For surfaces in R^m these facts are well known; however, the constructions leading to them are in large part inapplicable to spaces M^m. Hence approximations to by smooth equidistant (not necessarily convex) ones and normal polygonal paths, introduced (in the case of R³) by Yu. F. Borisov are used.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 66, pp. 114–132, 1976.     

Digital Jordan curves

A digital Jordan curve theorem is proved for a new topology defined on Z². This topology is compared with the classical Khalimsky and Marcus topologies used in digital topology. We show that the Jordan curves with respect to the topology defined, unlike the Jordan curves with respect to any of the two classical topologies mentioned, may turn at the acute angle . We also discuss a quotient topology of the new topology.     

Ein verbesserter Existenzsatz für Flächen konstanter mittlerer Krümmung

Let r be a recifiable closed Jordan curve in the euclidean 3-space IR³, and denote by A_r the infimum of the areas of all surfaces bounded by r. Then for every real number H with we show the existence of a surface with boundary curve r having constant mean curvature H (except in possible branching points). This improves a theorem of WENTE. Given an isolated minimal surface bounded by r for sufficiently small |H| we further prove the existence of a surface of constant mean curvature with boundary curve r which is close to the minimal surface.

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Diese Arbeit entstand aus dem ersten Teil meiner Dissertation, die auf Anregung und unter Anleitung von Prof. Dr. S. Hildebrandt geschrieben wurde. Ihm danke ich für seine ständige Unterstützung.     

Affine minimal hypersurfaces of rotation

We give a complete list of affine minimal surfaces inA ³ with Euclidean rotational symmetry, completing the treatise given in [1] and prove that these surfaces have maximal affine surface area within the class of all affine surfaces of rotation satisfying suitable boundary conditions. Besides we show that for rotationally symmetric locally strongly convex affine minimal hypersurfaces inA ⁿ ,n4, the second variation of the affine surface area is negative definite under certain conditions on the meridian.     

Singular curves bounding two unstable minimal disks

Let A⊆ℝ³ be a convex body and Γ the union of two Jordan curves on ∂A which meet each other at two points with prescribed angles. Then Γ bounds two unstable minimal disks. Received: 21 September 1998 / Revised version: 17 May 1999     

Integer points on curves and surfaces

Various upper bounds are given for the number of integer points on plane curves, on surfaces and hypersurfaces. We begin with a certain class of convex curves, we treat rather general surfaces in ³ which include algebraic surfaces with the exception of cylinders, and we go on to hypersurfaces in ⁿ with nonvanishing Gaussian curvature.Written with partial supports from NSF grant No. MCS-8211461.     

Eine Übertragung des Satzes von Bonnet auf Regelflächen im einfach isotropen Raum

J ₃ ⁽¹⁾ . Bonnet's classical theorem about ruled surfaces in the three-dimensional Euclidean space does not hold inJ ₃ ⁽¹⁾ . To get an isotropic version of this theorem the terms geodetic line and isogonal-trajectory of the generators are replaced by new, the isotropic space adapted properties of curves on ruled surfaces.     

An abstract formulation of variational refinement

In this paper, the theory of abstract splines is applied to the variational refinement of (periodic) curves that meet data to within convex sets in R^d. The analysis is relevant to each level of refinement (the limit curves are not considered here). The curves are characterized by an application of a separation theorem for multiple convex sets, and represented as the solution of an equation involving the dual of certain maps on an inner product space. Namely, Existence and uniqueness are established under certain conditions. The problem here is a generalization of that studied in (Kersey, Near-interpolatory subdivided curves, author's home page, 2003) to include arbitrary quadratic minimizing functionals, placed in the setting of abstract spline theory. The theory is specialized to the discretized thin beam and interval tension problems.     

Convexity preserving interpolation by algebraic curves and surfaces

The problem of interpolation by a convex curve to the vertices of a convex polygon is considered. A natural 1-parameter family ofC algebraic curves solving this problem is presented. This is extended to a solution, of a general Hermite-type problem, in, which the curve also interpolates to one or two prescribedtangents at any desired vertices of the polygon. The construction of these curves is a generalization of well known methods for generatingconic sections. Several properties of this family of algebraic curves are discussed. In addition, the method is generalized to convexC interpolation of strictly convex data sets inR ³ by algebraicsurfaces.     

Existence of Efficient Points in Vector Optimization and Generalized Bishop–Phelps Theorem

In a set without linear structure equipped with a preorder, we give a general existence result for efficient points. In a topological vector space equipped with a partial order induced by a closed convex cone with a bounded base, we prove another kind of existence result for efficient points; this result does not depend on the Zorn lemma. As applications, we study a solution problem in vector optimization and generalize the Bishop–Phelps theorem to a topological vector space setting by showing that the B-support points of any sequentially complete closed subset A of a topological vector space E is dense in A, where B is any bounded convex subset of E.     

Constant Minkowskian width in terms of double normals

We extend the notion of a double normal of a convex body from smooth, strictly convex Minkowski planes to arbitrary two-dimensional real, normed, linear spaces in two different ways. Then, for both of these ways, we obtain the following characterization theorem: a convex body K in a Minkowski plane is of constant Minkowskian width iff every chord I of K splits K into two compact convex sets K₁ and K₂ such that I is a Minkowskian double normal of K₁ or K₂. Furthermore, the Euclidean version of this theorem yields a new characterization of d-dimensional Euclidean ball where d 3.     

On large minimal blocking sets in PG(2,q)

The size of large minimal blocking sets is bounded by the Bruen–Thas upper bound. The bound is sharp when q is a square. Here the bound is improved if q is a non‐square. On the other hand, we present some constructions of reasonably large minimal blocking sets in planes of non‐prime order. The construction can be regarded as a generalization of Buekenhout's construction of unitals. For example, if q is a cube, then our construction gives minimal blocking sets of size q^4/3 + 1 or q^4/3 + 2. Density results for the spectrum of minimal blocking sets in Galois planes of non‐prime order is also presented. The most attractive case is when q is a square, where we show that there is a minimal blocking set for any size from the interval . © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 25–41, 2005.