Constant curved minimal 2-spheres in G(2,4)

This paper gives a complete classification for minimal 2-spheres with constant Gaussian curvature immersed in the complex Grassmann manifold G(2,4). Received: 14 May 1998 / Revised version: 12 October 1998     

Nonexistence of nonpositively curved surfaces¶with one embedded end

We prove certain complete nonpositively curved surfaces arising from general relativity isometrically immersible in R ³ do not exist, assuming square integrable second fundamental form. We provide an example showing the sharpness of our conditions. Received: 19 February 1999 / Revised version: 2 December 1999     

Towards the Willmore conjecture

We develop a variety of approaches, mainly using integral geometry, to proving that the integral of the square of the mean curvature of a torus immersed in must always take a value no less than . Our partial results, phrased mainly within the -formulation of the problem, are typically strongest when the Gauss curvature can be controlled in terms of extrinsic curvatures or when the torus enjoys further properties related to its distribution within the ambient space (see Sect. 3). Corollaries include a recent result of Ros [20] confirming the Willmore conjecture for surfaces invariant under the antipodal map, and a strengthening of the expected results for flat tori. The value arises in this work in a number of different ways – as the volume (or renormalised volume) of or , and in terms of the length of shortest nontrivial loops in subgroups of SO(4). Received April 26, 1999 / Accepted January 14, 2000 / Published online June 28, 2000     

A finite set of generators for the Kauffman bracket skein algebra

If F is a compact orientable surface it is known that the Kauffman bracket skein module of has a multiplicative structure. Our central result is the construction of a finite set of knots which generate the module as an algebra. We can then define an integer valued invariant of compact orientable 3-manifolds which characterizes . Received November 27, 1995; in final form September 29, 1997     

Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity

We show that a solution of the Cauchy problem for the KdV equation, has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x = 0. It is shown that for () data satisfying the condition the solution is analytic in both space and time variable. The above condition allows us to take as initial data the Dirac measure or the Cauchy principal value of 1/x. The argument is based on the recent progress on the well-posedness result by Bourgain [2] and Kenig-Ponce-Vega [20] and a systematic use of the dilation generator . Received 22 March 1999     

Linear embeddings of semialgebraic G-spaces

Let G be a compact semialgebraic linear group. We prove that every regular semialgebraic G-space admits a semialgebraic G-embedding into some semialgebraic orthogonal representation space of G. Received: 9 January 2001; in final form: 17 August 2001 / Published online: 28 February 2002     

Geometric diffeomorphism finiteness in low dimensions and homotopy group finiteness

The main results of this note consist in the following two geometric finiteness theorems for diffeomorphism types and homotopy groups of closed simply connected manifolds: 1. For any given numbers C and D the class of closed smooth simply connected manifolds of dimension which admit Riemannian metrics with sectional curvature bounded in absolute value by $\vert K \vert\le C$ and diameter bounded from above by D contains at most finitely many diffeomorphism types. In each dimension there exist counterexamples to the preceding statement. 2. For any given numbers C and D and any dimension m there exist for each natural number up to isomorphism always at most finitely many groups which can occur as the k-th homotopy group of a closed smooth simply connected m-manifold which admits a metric with sectional curvature and diameter . Received: 21 August 1999 / Accepted: 20 April 2001 / Published online: 19 October 2001     

Representing products of manifolds by edge-coloured graphs: the boundary case

In this paper we generalize the construction - introduced by Gagliardi and Grasselli in the closed case - of a coloured-graph representing the product of two manifolds, starting by two coloured graphs representing the manifolds themselves, to the boundary case. In particular we study the genus of the graph product of low dimensional manifold ( resp. n-spheres ) with m-disks. Received September 28, 1998; in final form January 5, 2000 / Published online October 11, 2000     

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     

Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves

We show that every unframed knot type in has a representative obtained by the Legendrian lifting of an immersed plane curve. This gives a positive answer to the question asked by V.I.Arnold in [3]. The Legendrian lifting lowers the framed version of the HOMFLY polynomial [20] to generic plane curves. We prove that the induced polynomial invariant can be completely defined in terms of plane curves only. Moreover it is a genuine, not Laurent, polynomial in the framing variable. This provides an estimate on the Bennequin-Tabachnikov number of a Legendrian knot. Received: 17 April 1996 / Revised: 12 May 1999 / Published online: 28 June 2000     

Analysis of a part design procedure

Summary. In this paper we analyze and illustrate a new "ab initio" part design procedure, in which, given a cost function which reflects performance, materials, and manufacturing considerations, the topology and the geometry of the part are automatically produced. The analysis is based on demonstration of, first, the compactness of the metric space over which the cost function is defined, and, second, lower semi-continuity of the cost function. Examples include beams and elastic supports. Received November 15, 1993     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As an application, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained by B. Palmer [Pa] and Y.L. Xin [Xin1] where they assume that the image of the Gauss map is bounded. We also prove a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional spaces. Received July 4, 1997 / Accepted October 9, 1997     

Computation of high order derivatives in optimal shape design

Summary. In shape optimization problems, each computation of the cost function by the finite element method leads to an expensive analysis. The use of the second order derivative can help to reduce the number of analyses. Fujii ([4], [10]) was the first to study this problem. J. Simon [19] gave the second order derivative for the Navier-Stokes problem, and the authors describe in [8], [11], a method which gives an intrinsic expression of the first and second order derivatives on the boundary of the involved domain. In this paper we study higher order derivatives. But one can ask the following questions: -- are they expensive to calculate? -- are they complicated to use? -- are they imprecise? -- are they useless? \medskip\noindent At first sight, the answer seems to be positive, but classical results of V. Strassen [20] and J. Morgenstern [13] tell us that the higher order derivatives are not expensive to calculate, and can be computed automatically. The purpose of this paper is to give an answer to the third question by proving that the higher order derivatives of a function can be computed with the same precision as the function itself. We prove also that the derivatives so computed are equal to the derivatives of the discrete problem (see Diagram 1). We call the discrete problem the finite dimensional problem processed by the computer. This result allows the use of automatic differentiation ([5], [6]), which works only on discrete problems. Furthermore, the computations of Taylor's expansions which are proposed at the end of this paper, could be a partial answer to the last question. Received January 27, 1993/Revised version received July 20, 1993     

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