Bicanonical pencil of a determinantal Barlow surface

In this paper, we study the bicanonical pencil of a Godeaux surface and of a determinantal Barlow surface. This study gives a simple proof for the unobstructedness of deformations of a determinantal Barlow surface. Then we compute the number of hyperelliptic curves in the bicanonical pencil of a determinantal Barlow surface via classical Prym theory.

     

Piecewise smooth developable surfaces

A.V. Pogorelov introduced developable surfaces with regularity (twice differentiability) violated along separate lines. In particular, the surface may not be smooth at all points of these lines (which form edges in this case). It is assumed that each point of the surface under consideration that belongs to a curvilinear edge (as well as any other interior point of this surface) has a neighborhood isometric to a Euclidean disk. In this paper we study the behavior of a developable surface near its curvilinear edge. It is proved that if two smooth pieces of a developable surface are adjacent along a curvilinear edge, then the spatial location of one of them in ?³ is uniquely determined by that of the other.     

Some curvature properties of complex surfaces

Summary In this paper, we are investigating curvature properties of complex two-dimensional Hermitian manifolds, particularly in the compact case. To do this, we start with the remark that the fundamental form of such a manifold is integrable, and we use the analogy with the locally conformal KÄhler manifolds, which follows from this remark. Among the obtained results, we have the following: a compact Hermitian surface for which either the Riemannian curvature tensor satisfies the KÄhler symmetries or the Hermitian curvature tensor satisfies the Riemannian Bianchi identity is KÄhler; a compact Hermitian surface of constant sectional curvature is a flat KÄhler surface; a compact Hermitian surface M with nonnegative nonidentical zero holomorphie Hermitian bisectional curvature has vanishing plurigenera, c₁(M) 0, and no exceptional curves; a compact Hermitian surface with distinguished metric, and positive integral Riemannian scalar curvature has vanishing plurigenera, etc.     

Simple factor dressing and the López–Ros deformation of minimal surfaces in Euclidean 3-space

The aim of this paper is to investigate a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a dressing on the conformal Gauss map can be defined. We study the induced transformation on minimal surfaces in the simplest case, the simple factor dressing, and show that the well-known López–Ros deformation of minimal surfaces is a special case of this transformation. We express the simple factor dressing and the López–Ros deformation explicitly in terms of the minimal surface and its conjugate surface. In particular, we can control periods and end behaviour of the simple factor dressing. This allows to construct new examples of doubly-periodic minimal surfaces arising as simple factor dressings of Scherk’s first surface.

     

On the capacity of surfaces in manifolds with nonnegative scalar curvature

Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at ∞. Even in the special case of ℝ³, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.     

On l-adic representations attached to Hilbert and Picard modular surfaces

In this paper we compute the l-adic Lie algebra of a product of l-adic representations associated to a Hilbert modular surface and a Picard modular surface.     

The zero surface tension limit two‐dimensional water waves

We consider two‐dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases. © 2005 Wiley Periodicals, Inc.     

Impedance profile inversion via the first transport equation

The subject of this paper is the inverse reflection problem for a stratified elastic half-space. That is, a linear elastic medium, whose elastic properties depend only on depth from a planar free surface, is stimulated at t = 0 by a plane wave impulsive source. The motion of a typical surface element is recorded for 0 ? t ? 2T. It is shown that this surface trace determines the acoustic impedance of the medium as a function of travel time, to (travel-time) depth T. Moreover, we give a precise characterization of those functions which may appear as surface traces, and show uniqueness, existence, and continuous dependence of the logarithm of the impedance as a function of the surface trace in the Sobolev H¹ topology.     

A CHARACTERIZATION OF EMBEDDABILITY OF GRAPHS ON SURFACES

In this article the Cn-graphs are introduced, by which a characterization of the embeddability of a graph on either an orientable surface or a non-orientable surface is provided.     

Double covers of Kummer surfaces

Besides its construction as a quotient of an abelian surface, a Kummer surface can be obtained as the quotient of a K3 surface by a -action. In this paper, we classify all such K3 surfaces. Our classification is expressed in terms of period lattices and extends Morrison’s criterion of K3 surfaces with a Shioda–Inose structure. Moreover, we list all the K3 surfaces associated to a general Kummer surface and provide very geometrical examples of this phenomenon.