Biharmonic maps in two dimensions

Biharmonic maps between surfaces are studied in this paper. We compute the bitension field of a map between surfaces with conformal metrics in complex coordinates. As applications, we show that a linear map from Euclidean plane into ${(\mathbb{R}^2, \sigma^2dwd \bar w)}$ is always biharmonic if the conformal factor σ is bianalytic; we construct a family of such σ, and we give a classification of linear biharmonic maps between 2 spheres minus a point. We also study biharmonic maps between surfaces with warped product metrics. This includes a classification of linear biharmonic maps between hyperbolic planes and some constructions of many proper biharmonic maps into a circular cone or a helicoid.     

A regularity result for polyharmonic maps with higher integrability

We prove a regularity result for critical points of the polyharmonic energy in with and p > 1. Our proof is based on a Gagliardo–Nirenberg-type estimate and avoids the moving frame technique. In view of the monotonicity formulae for stationary harmonic and biharmonic maps, we infer partial regularity in theses cases.     

Mannheim partner curves in 3-space

In this paper, we study Mannheim partner curves in three dimensional space. We obtain the necessary and sufficient conditions for the Mannheim partner curves in Euclidean space and Minkowski space , respectively. Some examples are also given. Supported by NSFC, No.10371013; Joint Research of NSFC and KOSEF, NEU     

Constrained Willmore surfaces

Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the critical points of arbitrary geometric functionals on the space of immersions under the constraint that the admissible variations infinitesimally preserve the conformal structure. Besides constrained Willmore surfaces we discuss in some detail examples of constrained minimal and volume critical surfaces, the critical points of the area and enclosed volume functional under the conformal constraint. C. Bohle, G. P. Peters and U. Pinkall are partially supported by DFG SPP 1154.     

Conformal immersions of warped products

We prove a decomposition theorem for conformal immersions \(f\colon\;M^n\to {\mathbb{R}}^{N}\) into Euclidean space of a warped product of Riemannian manifolds \(M^n:=M_0\times_\rho\Pi_{i=1}^k M_i\) of dimension n ≥ 3 under the assumption that the second fundamental form \(\alpha \colon TM \times TM\to T^\perp M\) of f satisfies \(\alpha|_{TM_i\times TM_j}=0\) for i ≠ j. It generalizes the corresponding theorem of Nölker for isometric immersions as well as our previous result on conformal immersions of Riemannian products. In particular, we determine all conformal representations of Euclidean space of dimension n ≥ 3 as a warped product of Riemannian manifolds. As a consequence, we classify the conformally flat warped products.     

The stress-energy tensor for biharmonic maps

Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to construct new examples of biharmonic maps, then classify maps with vanishing or parallel stress-energy tensor and Riemannian immersions whose stress-energy tensor is proportional to the metric, thus obtaining a weaker but high-dimensional version of the Hopf Theorem on compact constant mean curvature immersions. We also relate the stress-energy tensor of the inclusion of a submanifold in Euclidean space with the harmonic stress-energy tensor of its Gauss map. S. Montaldo was supported by PRIN-2005 (Italy): Riemannian Metrics and Differentiable Manifolds. C. Oniciuc was supported by a CNR-NATO (Italy) fellowship and by the Grant CEEX, ET, 5871/2006 (Romania).     

Examples of area-minimizing surfaces in the sub-Riemannian Heisenberg group $${\mathbb{H}^1}$$ with low regularity

We give new examples of entire area-minimizing t-graphs in the sub-Riemannian Heisenberg group . They are locally Lipschitz in Euclidean sense. Some regular examples have prescribed singular set consisting of either a horizontal line or a finite number of horizontal halflines extending from a given point. Amongst them, a large family of area-minimizing cones is obtained. Research supported by MEC-Feder grant MTM2007-61919.     

Blaschke’s problem for hypersurfaces

We solve Blaschke’s problem for hypersurfaces of dimension . Namely, we determine all pairs of Euclidean hypersurfaces that induce conformal metrics on M ⁿ and envelop a common sphere congruence in .     

Global Representation of Harmonic and Biharmonic Functions

Using some well-known properties of harmonic functions, such as the maximum principle and the existence of the Dirichlet solution, certain representation theorems for harmonic and biharmonic functions are proved in .     

Surfaces in $$\mathbb {R}^7$$ obtained from harmonic maps in $$S^6$$S6.

We will investigate the local geometry of the surfaces in the 7-dimensional Euclidean space associated to harmonic maps from a Riemann surface \(\varSigma \) into \(S^6\). By applying methods based on the use of harmonic sequences, we will characterize the conformal harmonic immersions \(\varphi :\varSigma \rightarrow S^6\) whose associated immersions \(F:\varSigma \rightarrow \mathbb {R}^7\) belong to certain remarkable classes of surfaces, namely: minimal surfaces in hyperspheres; surfaces with parallel mean curvature vector field; pseudo-umbilical surfaces; isotropic surfaces.