Discrete curve theory in space forms: planar elastic and area-constrained elastic curves

In this paper, we give explicit parametrizations for Bour type surfaces in various three-dimensional space forms, using Weierstrass-type representations. We also determine classes and degrees of some Bour type zero mean curvature surfaces in three-dimensional Minkowski space.

A Hopf hypersurface in a (para-)Kaehler manifold is a real hypersurface for which one of the principal directions of the second fundamental form is the (para-)complex dual of the normal vector. We consider particular Hopf hypersurfaces in the space of oriented geodesics of a non-flat space form of dimension greater than 2. For spherical and hyperbolic space forms, the oriented geodesic space admits a canonical Kaehler-Einstein and para-Kaehler-Einstein structure, respectively, so that a natural notion of a Hopf hypersurface exists. The particular hypersurfaces considered are formed by the oriented geodesics that are tangent to a given convex hypersurface in the underlying space form. We prove that a tangent hypersurface is Hopf in the space of oriented geodesics with respect to this canonical (para-)Kaehler structure iff the underlying convex hypersurface is totally umbilic and non-flat. In the case of 3 dimensional space forms, however, there exists a second canonical complex structure which can also be used to define Hopf hypersurfaces. We prove that in this dimension, the tangent hypersurface of a convex hypersurface in the space form is always Hopf with respect to this second complex structure.

Curve straightening and a minimax argument for closed elastic curves

In this paper we consider two special classes of constrained Willmore tori in the 3-sphere. The first class is given by the rotation of closed elastic curves in the upper half plane - viewed as the hyperbolic plane - around the x-axis. The second is given as the preimage of closed constrained elastic curves, i.e., elastic curve with enclosed area constraint, in the round 2-sphere under the Hopf fibration. We show that all conformal types can be isometrically immersed into S^3 as constrained Willmore (Hopf) tori and write down all constrained elastic curves in H^2 and S^2 in terms of the Weierstrass elliptic functions. Further, we determine the closing condition for the curves and compute the Willmore energy and the conformal type of the resulting tori.

We propose a discrete surface theory in ℝ3 that unites the most prevalent versions of discrete special parametrizations. Our theory encapsulates a large class of discrete surfaces given by a Lax representation and, in particular, the one-parameter associated families of constant curvature surfaces. Our theory is not restricted to integrable geometries, but extends to a general surface theory. A quad net is a map from a strongly regular polytopal cell decomposition of a surface with all faces being quadrilaterals into ℝ3 with nonvanishing straight edges. A polytopal cell decomposition is strongly regular if each edge connects distinct vertices and meets at most two faces. Notice, in particular, that nonplanar faces are admissible. In discrete differential geometry, quad nets are understood as discretizations of parametrized surfaces [9, 10, 13]. In this agenda many classes of special surfaces have been discretized using algebro-geometric approaches for integrable geometry—originally using discrete analogues of soliton theory techniques (e.g., discrete Lax pairs and finite-gap integration [6]) to construct nets, but more recently using the notion of 3D consistency (reviewed in [10]). As in the smooth setting, these approaches have been successfully applied to space forms (see, e.g., [15–17, 22]). As an example consider the case of K-surfaces, i.e., surfaces of constant negative Gauß curvature. The integrability equations of classical surface theory are equivalent to the famous sine-Gordon equation [1, 20]. In an integrable discretization, the sine-Gordon equation becomes a finite difference equation for which integrability is encoded by a certain closing condition around a 3D cube. Both in the smooth and discrete settings, integrability is bound to specific choices of parameterizations, such as asymptotic line parametrizations for K-surfaces. In this way different classes of surfaces, such as minimal surfaces or surfaces of constant mean curvature (CMC), lead to different partial differential equations and give rise to different parametrizations. In the discrete case, this is reflected by developments that treat different special surfaces by disparate approaches. These integrable discretizations maintain characteristic properties of their smooth counterparts (e.g., the transformation theory of Darboux, Bäcklund, Bianchi, etc.). What has been lacking, however, is a unified discrete theory that lifts the restriction of special surface parametrizations. Indeed, different from the case of classical smooth surface theory, existing literature does not provide a general discrete theory for quad nets.

We give a characterization of isoparametric surfaces in three dimensional space forms with the help of minimal surfaces. Let us consider a surface in a three dimensional space form with the following property: through each point of the surface pass three curves such that the ruled orthogonal surface determined by each of them is minimal. We prove that necessarily the initial surface is isoparametric. It is also shown, that the curves are necessarily geodesics. On other hand, using the classification of isoparametric surfaces it is possible to prove that they have the above property along every geodesic. So, we have a characterization. As a preliminary result we prove that a surface with two geodesics, through every point, which are helices of the ambient, has parallel shape operator.

A note on equivariant biharmonic maps and stable biharmonic maps

In this paper, we find a full description of concircular hypersurfaces in space forms as a special family of ruled hypersurfaces. We also characterize concircular helices in 3-dimensional space forms by means of a differential equation involving the concircular factor and their curvature and torsion, and we show that the concircular helices are precisely the geodesics of the concircular surfaces.

Constant mean curvature hypersurfaces with constant angle in semi-Riemannian space forms

Let x: M → R n+p (c) be an n-dimensional compact, possibly with bound- ary, submanifold in an (n + p)-dimensional space form R n+p (c). Assume that r is even and r ∈{ 0,1, ... ,n − 1}, in this paper we introduce rth mean curvature function Sr and (r + 1)-th mean curvature vector field Sr+1 .W e callM to be an r-minimal submanifold if Sr+1 ≡ 0o nM, we note that the concept of 0-minimal submani- fold is the concept of minimal submanifold. In this paper, we define a functional Jr(x) = � M Fr(S0,S2, ... ,Sr)dv of x: M → R n+p (c), by calculation of the first varia- tional formula of Jr we show that x is a critical point of Jr if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We cal- culate the second variational formula of Jr and prove that there exists no compact without boundary stable r-minimal submanifold with Sr > 0 in the unit sphere S n+p . When r = 0, noting S0 = 1, our result reduces to Simons' result: there exists no compact without boundary stable minimal submanifold in the unit sphere S n+p .

The symplectic structure of curves in three dimensional spaces of constant curvature and the equations of mathematical physics

In the three dimensional Riemannian space forms, we introduce a natural moving frame to define associate curve of a curve. Using the notion of associate curve we give a new necessary and sufficient condition of which a Frenet curve is a Mannheim curve or Mannheim partner curve in the three dimensional Euclidean space. Then we generalize these conclusions to the curves which lie on the three dimensional Riemannian sphere and the curves which lie in the three dimensional hyperbolic space. We also give the geometric characterizations of these curves. Our methods can be easily used to reveal the properties of the curves on space forms.

An Immersion of an $n$-dimensional Real Space Form into an $n$-dimensional Complex Space Form

We give a necessary and sufficient condition for the existence of spacelike minimal surfaces in $4$-dimensional Lorentzian space forms, which is a generalization of the Ricci condition for minimal surfaces in $3$-dimensional Riemannian space forms.

Let X N ( c ) {X^N}(c) denote the N-dimensional simply connected space form of constant curvature c. We consider a problem to classify those minimal surfaces in X N ( c ) {X^N}(c) which are locally isometric to minimal surfaces in X 3 ( c ) {X^3}(c) . In this paper we solve this problem in the case where N = 4 N = 4 , and give a result also in higher codimensional cases.