Ribaucour transformations for flat surfaces in the hyperbolic 3-space

We consider Ribaucour transformations for flat surfaces in the hyperbolic 3-space, H³${\mathbb{H}}^3$. We show that such transformations produce complete, embedded ends of horosphere type and curves of singularities which generically are cuspidal edges. Moreover, we prove that these ends and curves of singularities do not intersect. We apply Ribaucour transformations to rotational flat surfaces in H³${\mathbb{H}}^3$ providing new families of explicitly given flat surfaces H³${\mathbb{H}}^3$ which are determined by several parameters. For special choices of the parameters, we get surfaces that are periodic in one variable and surfaces with any even number or an infinite number of embedded ends of horosphere type.     

Flat slant surfaces in complex projective and complex hyperbolic planes

For surfaces in complex space forms with almost complex structure J, flat surfaces are the simplest ones from intrinsic point of view. From J-action point of view, the most natural surfaces are slant surfaces. The classification of flat slant surfaces in C² was done in [2]. In this paper we apply a result of [5] to study flat slant surfaces in CP ² and CH ². We prove that, for any θ, there exist infinitely many flat θ-slant surfaces in CP ² and CH ². And there does not exist flat half-minimal proper slant surface in CP ² and in CH ².     

On the generalized Weyl problem for flat metrics in the hyperbolic 3-space

The paper deals with the study of complete embedded flat surfaces in H³${\mathbb{H}}^3$ with a finite number of isolated singularities. We give a detailed information about its topology, conformal type and metric properties. We show how to solve the generalized Weyl?s problem of realizing isometrically any complete flat metric with Euclidean singularities in H³${\mathbb{H}}^3$ which gives the existence of complete embedded flat surfaces with a finite arbitrary number of isolated singularities.     

Four Classes of Surfaces with Constant Mean Curvature in S 3 × R and H 3 × R

Deforming rotation surfaces with constant mean curvature in S ³ and H ³ to S ³ × R and H ³ × R respectvely, we give four classes of surfaces with mean curvature vector of constant length in S ³ × R and H ³ × R. We have complete minimal surfaces in S ³ × R and H ³ × R. Also we obtain minimal 2-tori in S ³ × S ¹, some of which are embedded.     

The Cauchy problem for the Liouville equation and Bryant surfaces

We give a construction that connects the Cauchy problem for the 2-dimensional elliptic Liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H³, and solve both of them. We construct the unique mean curvature one surface in H³ that passes through a given curve with a given unit normal along it, and provide diverse applications. In particular, topics such as period problems, symmetries, finite total curvature, planar geodesics, rigidity, etc. are treated for these surfaces.     

Extrinsic diameter of immersed flat tori in <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

Enomoto, Weiner and the first author showed the rigidity of the Clifford torus amongst the class of embedded flat tori in S ³. In the proof of that result, an estimate of extrinsic diameter of flat tori plays a crucial role. It is reasonable to expect that the same rigidity holds in the class of immersed flat tori in S ³. In this paper, we give a new method for characterizing immersed flat tori in S ³ with extrinsic diameter π, which is a somewhat similar technique to the proof of the 6-vertex theorem for certain closed plane curves given by the second author. As an application, we show that the Clifford torus is rigid in the class of immersed flat tori whose mean curvature functions do not change sign. Recently, the global behaviour of flat surfaces in H ³ and R ³ regarded as wave fronts has been studied. We also give here a formulation of flat tori in S ³ as wave fronts. As an application, we shall exhibit a flat torus as a wave front whose extrinsic diameter is less than π.     

Universal moduli spaces of surfaces with flat bundles and cobordism theory

For a compact, connected Lie group G, we study the moduli of pairs (Σ,E), where Σ is a genus g Riemann surface and E→Σ is a flat G-bundle. Varying both the Riemann surface Σ and the flat bundle leads to a moduli space , parametrizing families Riemann surfaces with flat G-bundles. We show that there is a stable range in which the homology of is independent of g. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range, in terms of the homology of an explicit infinite loop space. Rationally, the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller κ-classes, and the ring of characteristic classes of principal G-bundles, H^∗(BG). Equivalently, our theorem calculates the homology of the moduli space of semi-stable holomorphic bundles on Riemann surfaces.We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.     

Projectively flat affine surfaces

We study non-degenerate affine surfaces in A³ with a projectively flat induced connection. The curvature of the affine metric , the affine mean curvature H, and the Pick invariant J are related by . Depending on the rank of the span of the gradients of these functions, a local classification of three groups is given. The main result is the characterization of the projectively flat but not locally symmetric surfaces as a solution of a system of ODEs. In the final part, we classify projectively flat and locally symmetric affine translation surfaces.     

Equidistant hypersurfaces of the bidisk

The following are notes on the geometry of the bidisk, H ² × H ². In particular, we examine the properties of equidistant surfaces in the bidisk.     

On the unique continuation problem for CR mappings into nonminimal hypersurfaces

We prove the following results on the unique continuation problem for CR mappings between real smooth hypersurfaces in ? ⁿ . If the CR mappingH extends holomorphically to one side of the source manifoldM near the pointp ₀ εM, the target manifoldM′ contains a holomorphic hypersurface σ′ throughp′₀ =H(p ₀ (i.e.,M′ is nonminimal atp′ ₀), andH(M) ? Σ′ (forcingM to be nonminimal atp ₀), then the transversal component ofH is not flat atp ₀. Furthermore, we show that the assumption thatH extends holomorphically to one side ofM cannot be removed in general. Indeed, we give an example of a smooth CR mappingH, withM, M′ ? ?², real analytic and of infinite type atp ₀ andp′₀ respectively (without being Levi flat), such thatH(M) ? Σ′ but the transversal component ofH is flat atp ₀ (in particular,H is not real analytic!). However, we show that ifM andM′ are assumed to be real analytic, and if the sourceM is “sufficiently far from being Levi flat” in a certain sense (so as to exclude the above mentioned counterexample) then the assumption thatH extends holomorphically to one side ofM can be dropped. Also, in the general case, we prove that the rate of vanishing of the transversal component cannot be too rapid (unlessH(M) ? Σ′), and we relate the possible rate of vanishing to the order of vanishing of the Levi form on a certain holomorphic submanifold ofM.     

Hypersurfaces in Hn and the space of its horospheres

A classical theorem, mainly due to Aleksandrov [Al2] and Pogorelov [P], states that any Riemannian metric on S ² with curvature K > —1 is induced on a unique convex surface in H ³ . A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon happens with yet another metric on immersed surfaces, which we call the horospherical metric.?This result extends in higher dimensions, the metrics obtained are then conformally flat. One can also study equivariant immersions of surfaces or the metrics obtained on the boundaries of hyperbolic 3-manifolds. Some statements which are difficult or only conjectured for the induced metric or the third fundamental form become fairly easy when one considers the horospherical metric, which thus provides a good boundary condition for the construction of hyperbolic metrics on a manifold with boundary.?The results concerning the third fundamental form are obtained using a duality between H ³ and the de Sitter space . In the same way, the results concerning the horospherical metric are proved through a duality between H ⁿ and the space of its horospheres, which is naturally endowed with a fairly rich geometrical structure. Submitted: March 2001, Revised: November 2001.     

A characterization of constant mean curvature surfaces in homogeneous 3-manifolds

It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this paper we describe all the surfaces with holomorphic Hopf differential in the homogeneous 3-manifolds isometric to H²×R or having isometry group isomorphic either to the one of the universal cover of PSL(2,R), or to the one of a certain class of Berger spheres. It turns out that, except for the case of these Berger spheres, there exist some exceptional surfaces with holomorphic Hopf differential and non-constant mean curvature.     

Stationary rotating surfaces in Euclidean space

A stationary rotating surface is a compact surface in Euclidean space whose mean curvature H at each point x satisfies 2H(x) = a r(x)² + b, where r(x) denotes the distance from x to a fixed straight-line L, and a and b are constants. These surfaces are solutions of a variational problem that describes the shape of a drop of incompressible fluid in equilibrium by the action of surface tension when it rotates about L with constant angular velocity. The effect of gravity is neglected. In this paper we study the geometric configurations of such surfaces, focusing the relationship between the geometry of the surface and the one of its boundary. As special cases, we will consider two families of such surfaces: axisymmetric surfaces and embedded surfaces with planar boundary.     

The prescribed mean curvature equation for nonparametric surfaces

We study the prescribed mean curvature equation for nonparametric surfaces, obtaining existence and uniqueness results in the Sobolev space W^2,p. We also prove that under appropriate conditions the set of surfaces of mean curvature H is a connected subset of W^2,p. Moreover, we obtain existence results for a boundary value problem which generalizes the one-dimensional periodic problem for the mean curvature equation.     

Flat Surfaces in \mathbb{L}4

We give a global conformal representation for flat surfaces with a flat normal bundle in the standard flat Lorentzian space form ⁴. Particularly, flat surfaces in hyperbolic 3-space, the de Sitter 3-space, the null cone, and other numerous examples aredescribed.     

A complex hyperbolic structure for moduli of cubic surfaces

We show that the moduli space M of marked cubic surfaces is biholomorphic to (B⁴ − H)/Г, where B⁴ is complex hyperbolic four-space, Γ is a specific group generated by complex reflections, and H is the union of reflection hyperplanes for Γ. Thus M has a complex hyperbolic structure, i.e., an (incomplete) metric of constant negative holomorphic sectional curvature.     

Brauer groups and Amitsur cohomology for general commutative ring extensions

Exact couples are interconnected families of long exact sequences extending the short exact sequences usually derived from spectral sequences. This is exploited to give a long exact sequence connecting Amitsur cohomology groups $\text{H>}{}^{\mathrm{n}}\text{(}\text{S}\text{R}\text{, U)}$ (where U means the multiplicative group) and $\text{H}{}^{\mathrm{n}}\text{(}\text{S}\text{R}\text{, Pic)}$ and a third sequence of groups Hⁿ(J), for every faithfully flat commutative R-algebra S. This same sequence is derived in another way without assuming faithful flatness and Hⁿ(J) is identified explicitly as a certain subquotient of a group of isomorphism classes of pairs (P, α) with P a rank one, projective Sⁿ-module and α an isomorphism from the coboundary of P (inPicS^{n + 1}) toS^{n + 1}. (Here Sⁿ denotes repeated tensor product of S over R.) This last formulation allows us to construct a homomorphism of the relative Brauer group $\text{B(}\text{S}\text{R}\text{)}$ to H²(J) which is a monomorphism when S is faithfully flat over R, and an isomorphism when some S-module is faithfully projective over R. The first approach also identifies H²(J) with Ker[H²(R, U)→H²(S, U)], where H²(R, U) denotes the ordinary, Grothendieck cohomology (in the étale topology, for example).     

Special Slant Surfaces and a Basic Inequality

A slant immersion is an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. A slant submanifold is called proper if it is neither holomorphic nor totally real. In [2], the author proved that, for any proper slant surface M with slant angle θ in a complex-space-form $?detilde M^2(4?silon)$ with constant holomorphic sectional curvature 4?, the squared mean curvature and the Gauss curvature of M satisfy the following basic inequality: H²(p) ≥ 2K(p) ? 2(1 + 3 cos²θ)?. Every proper slant surface satisfying the equality case of this inequality is special slant. One purpose of this article is to completely classify proper slant surfaces which satisfy the equality case of this inequality. Another purpose of this article is to completely classify special slant surfaces with constant mean curvature. Further results on special slant surfaces are also presented.     

Helicoidal surfaces under the cubic screw motion in Minkowski 3-space

In this paper, we construct helicoidal surfaces under the cubic screw motion with prescribed mean or Gauss curvature in Minkowski 3-space . We also find explicitly the relation between the mean curvature and Gauss curvature of them. Furthermore, we discuss helicoidal surfaces under the cubic screw motion with H²=K and prove that these surfaces have equal constant principal curvatures.