The generalized Weierstrass representation is used to analyze the asymptotic behavior of a constant mean curvature surface that arises locally from an ordinary differential equation (ODE) with a regular singularity. We prove that a holomorphic perturbation of an ODE that represents a Delaunay surface generates a constant mean curvature surface which has a properly immersed end that is asymptotically Delaunay. Furthermore, that end is embedded if the Delaunay surface is unduloidal.

An extension of a result of Sela shows that if Γ is a torsion-free word hyperbolic group, then the only homomorphisms Γ→Γ with finite-index image are the automorphisms. It follows from this result and properties of quasiregular mappings, that if M is a closed Riemannian n-manifold with negative sectional curvature (), then every quasiregular mapping f:M→M is a homeomorphism. In the constant-curvature case the dimension restriction is not necessary and Mostow rigidity implies that f is homotopic to an isometry. This is to be contrasted with the fact that every such manifold admits a non-homeomorphic light open self-mapping. We present similar results for more general quotients of hyperbolic space and quasiregular mappings between them. For instance, we establish that besides covering projections there are no π1-injective proper quasiregular mappings f:M→N between hyperbolic 3-manifolds M and N with non-elementary fundamental group.

We characterize least-perimeter enclosures of prescribed area on some piecewise smooth manifolds, including certain polyhedra, double spherical caps, and cylindrical cans.

In this paper we prove a uniqueness theorem for minimal discs in R3 spanning a polygonal boundary.

A minimal surface is a surface with vanishing mean curvature. In this paper we study self θ -congruent minimal surfaces, that is, surfaces which are congruent to their θ-associates under rigid motions in R3 for 0 ≤ θ < 2π. We give necessary and sufficient conditions in terms of its Weierstrass pair for a surface to be self θ-congruent. We also construct some examples and give an application.

In connection with the thesis of Ch. Charitos (1989), T. Hasanis posed the following question.

The classical notion of a two-dimensional develpable surface in Euclidean three-space is extended to the case of arbitrary dimension and codimension. A collection of characteristic properties is presented. The theorems are stated with the minimal possible integer smoothness. The main tool of the investigation is Cartan's moving frame method.

In this paper we describe the moduli spaces of degree d branched superminimal immersions of a compact Riemann surface of genus g into S4. We prove that when d ≥ max {2g, g + 2}, such spaces have the structure of projectivzed fibre products and are path-connected quasi projective varieties of dimension 2d − g + 4. This generalizes known results for spaces of harmonic 2-spheres in S4.

Let γ:x = x(u) for a≤ u ≤ b be a closed curve in n dimensional euclidean space En (for n ≥ 2) referred to some point P0, which does not lie on γ, as origin. We suppose that γ is smooth in the sense that the cartesian coordinates are class C2 functions of the parameter u, and that dx/du is non-vanishing so that a tangent vector is everywhere well defined. These properties are also assumed to hold in an obvious way at the join of the end points u = a and u = b. As P moves on γ its position vector x(u) intersects the surface of the unit sphere centred at P0 in a closed curve γ0. Note that γ0: x = x0(u) may not be smooth everywhere. If there are points P on γ where P0P is tangential to γ at P then dx0/du =0 at the corresponding point on γ0, and γ0 may have a cusp there. We assume that γ0 is smooth except at a finite number of points. We define the total swing of the position vector to γ to be the arc length L0 of γ0. Clearly L0 is not an invariant but depends on the choice of the origin P0 in relation to γ. The total (first) curvature of γ is an invariant and is defined by

where s is the arc length on γ, K(S) is the curvature and ‖v‖ is the euclidean norm of the vector v. Note that LT is also the arc length of the closed curve γ1, described on a unit sphere by the unit tangent t = dx/ds to γ as position vector, with the centre of the sphere as origin, γ1 is the spherical indicatrix of t. Our purpose is to establish the result

for all choices of P0.

We consider hypersurfaces of En+1 whose position vector x satisfies Δx = Ax + B, where Δ is the induced Laplacian, and prove that these are open parts of minimal hypersurfaces, hyperspheres or generalized circular cylinders.

In this paper we consider O. Bonnet III-isometry (or BIII-isometry) of surfaces in 3-dimensional Euclidean space E3 Suppose a map F: M → M* is a diffeomorphism, and F* (III*) = III, ki(m) = k*i (m*), i = 1, 2, where m ∈ M, m* ∈ M*, m* = F (m), ki and k*i are the principal curvatures of surfaces M and M* at the points m and m*, respectively, III and III* are the third fundmental forms of M and M*, respectively. In this case, we call F an O. Bonnet III-isometry from M to M*. O. Bonnet I-isometries were considered in references [1]-[5].

We distinguish three cases about BIII-surfaces, which admits a non-trivial BIII-ismetry. We obtain some geometric properties of BIII-surfaces and BIII-isometries in these three cases; see Theorems 1, 2, 3 (in Section 2). We study some special BIII-surfaces: the minimal BIII-surfaces; BIII-surfaces of revolution; and BIII-surfaces with constant Gaussian curvature; see Theorems 4, 5, 6 (in Section 3).

Simply connected conformally flat Riemannian manifolds are characterized as hypersurfaces in the light cone of a standard flat Lorentzian space, transversal to its generators. Some applications of this fact are given.

This paper introduces a tensor that contains the Riemannian curvature tensor and the conformal curvature tensor as special examples in the Riemannian space (Mn, g), and by using this tensor we define C-semi-symmetric space. In this paper, we have the following main result: if there is a non-trivial concircular transformation between two C-semi-symmetric spaces, then both spaces are of quasi-constant curvature.

In this paper we consider how much we can say about an irreducible symmetric space M which admits a single hypersurface with at most two distinct principal curvatures. Then we prove that if N is conformally flat, then N is quasiumbilical and M must be a sphere, a real projective space or the noncompact dual of a sphere or a real projective space.

Let f and g denote immersions of the n-manifolds M and N, respectively, in Rn+1. We say that f is athwart to g if f(M) and g(N)m have no tangent hyperplane in common. In this paper necessary conditions for athwartness are obtained.