Biminimal hypersurfaces in a sphere

In this paper, we mainly concentrate on the biminimal hypersurfaces in a sphere. First, we obtain some rigidity theorems for biminimal hypersurfaces. Then we give a classification of non-minimal biminimal isoparametric hypersurfaces in a sphere.     

Stability of twistor lifts for surfaces in four-dimensional manifolds as harmonic sections

We prove that the twistor lifts of certain twistor holomorphic surfaces in four-dimensional manifolds are weakly stable harmonic sections. As a corollary, if ambient spaces are self-dual Einstein manifolds with nonnegative scalar curvature, then the twistor lifts of twistor holomorphic surfaces are weakly stable. Moreover, for certain surfaces in four-dimensional hyperkähler manifolds, we show that the surfaces are twistor holomorphic if their twistor lifts are weakly stable harmonic sections. In particular, we characterize twistor holomorphic surfaces in four-dimensional Euclidean space by weak stability of the twistor lifts.     

Symmetries of sub-Riemannian surfaces

We obtain some results on symmetries of sub-Riemannian surfaces. In case of a contact sub-Riemannian surface we base on invariants found by Hughen [15]. Using these invariants, we find conditions under which a sub-Riemannian surface does not admit symmetries. If a surface admits symmetries, we show how invariants help to find them. It is worth noting, that the obtained conditions can be explicitly checked for a given contact sub-Riemannian surface. Also, we consider sub-Riemannian surfaces which are not contact and find their invariants along the surface where the distribution fails to be contact.     

Bernstein theorems for space-like graphs with parallel mean curvature and controlled growth

In this paper, we obtain an Ecker–Huisken-type result for entire space-like graphs with parallel mean curvature.     

Some constructions of biharmonic maps and Chen’s conjecture on biharmonic hypersurfaces

We give several construction methods and use them to produce many examples of proper biharmonic maps including biharmonic tori of any dimension in Euclidean spheres (Theorem 2.2, Corollary 2.3, Corollary 2.4 and Corollary 2.6), biharmonic maps between spheres (Theorem 2.9) and into spheres (Theorem 2.10) via orthogonal multiplications and eigenmaps. We also study biharmonic graphs of maps, derive the equation for a function whose graph is a biharmonic hypersurface in a Euclidean space, and give an equivalent formulation of Chen’s conjecture on biharmonic hypersurfaces by using the biharmonic graph equation (Theorem 4.1) which paves a way for the analytic study of the conjecture.     

On the geometrized Skyrme and Faddeev models

The higher power derivative terms involved in both Faddeev and Skyrme energy functionals correspond to _σ2${\sigma }_2$-energy, introduced by Eells and Sampson (1964) [1]. This paper provides a detailed study of the first and second variation formulae associated to this energy. Some classes of (stable) critical points are outlined.     

Minimal hypersurfaces in a nearly Kaehler 6-sphere

In this paper we show that for a compact minimal hypersurface M$M$ of constant scalar curvature in the unit sphere S⁶$S^6$ with the shape operator A$A$ satisfying ‖A‖²>5${\Vert A\Vert }^2>5$, there exists an eigenvalue λ>10$\lambda >10$ of the Laplace operator of the hypersurface M$M$ such that ‖A‖²=λ−5${\Vert A\Vert }^2=\lambda -5$. This gives the next discrete value of ‖A‖²${\Vert A\Vert }^2$ greater than 0 and 5.     

The Gauss map of minimal surfaces in the Anti-de Sitter space

It is proved that a pair of spinors satisfying a Dirac-type equation represent surfaces immersed in Anti-de Sitter space with prescribed mean curvature. Here, we consider Anti-de Sitter space as the Lie group SU_1,1${\text{<small-caps>SU</small-caps>}}_{1,1}$ endowed with a one-parameter family of left-invariant metrics where only one of them is bi-invariant and corresponds to the isometric embedding of Anti-de Sitter space as a quadric in R^2,2${\mathbb{R}}^{2,2}$. We prove that the Gauss map of a minimal surface immersed in SU_1,1${\text{<small-caps>SU</small-caps>}}_{1,1}$ is harmonic. Conversely, we exhibit a representation of minimal surfaces in Anti-de Sitter space in terms of a given harmonic map.     

Homogeneous Finsler spaces of negative curvature

We prove that a homogeneous Finsler space with non-positive flag curvature and strictly negative Ricci scalar is a simply connected manifold.     

Harmonic morphisms and bicomplex manifolds

We use functions of a bicomplex variable to unify the existing constructions of harmonic morphisms from a 3-dimensional Euclidean or pseudo-Euclidean space to a Riemannian or Lorentzian surface. This is done by using the notion of complex-harmonic morphism between complex-Riemannian manifolds and showing how these are given by bicomplex-holomorphic functions when the codomain is one-bicomplex dimensional. By taking real slices, we recover well-known compactifications for the three possible real cases. On the way, we discuss some interesting conformal compactifications of complex-Riemannian manifolds by interpreting them as bicomplex manifolds.     

On spherically symmetric Finsler metrics of scalar curvature

Spherically symmetric Finsler metrics form a rich class of Finsler metrics. In this paper we find equations that characterize spherically symmetric Finsler metrics of scalar flag curvature. By using these equations, we construct infinitely many non-projectively flat spherically symmetric Finsler metrics of scalar curvature.     

On time-like Willmore surfaces in Minkowski space

We study time-like surfaces in Minkowski space, which are critical points of the Willmore energy. Transforming the fourth order Willmore equation into a quasi-linear, second order hyperbolic system, we prove existence, uniqueness and symmetry properties of such surfaces, subject to geometric initial conditions.     

Spacelike surfaces with positive definite second fundamental form in 3D spacetimes

For a spacelike surface with positive definite second fundamental form in any 3-dimensional Lorentzian manifold, a new formula relating its mean and Gauss curvature with the Gauss curvature of the second fundamental form is obtained. As an application, necessary and sufficient conditions are established in order to prove that such a compact spacelike surface is totally umbilical.