Weierstrass Type Representation of Willmore Surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we reformulate the Euler-Lagrange equations of Willmore surfaces in S^n as the flatness of a family of certain loop algebra-valued 1-forms. Therefore we can give the Weierstrass type representation of conformal Willmore surfaces. We also discuss the relations between conformal Willmore surfaces in S^n and minimal surfaces in constant curvature spaces S^n, R^n, H^n, and prove that some special Willmore surfaces can be derived from minimal surfaces in S^n, R^n, H^n.     

A characterization of the Ejiri torus in <Emphasis Type="Italic">S</Emphasis><Superscript>5</Superscript>

We conjecture that a Willmore torus having Willmore functional between 2π ² and 2π ² \(\sqrt 3 \) is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri’s torus in S ⁵ is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in S ⁿ by reducing them into elastic curves in S ³, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S ⁵ attains the minimum 2π ² \(\sqrt 3 \), which indicates our conjecture holds true for Willmore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S ⁵. Moreover, similar to Li and Vrancken, we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves in S ³. All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough.     

Conformal geometry of surfaces in Lorentzian space forms

We study the conformal geometry of an oriented space-like surface in three-dimensional Lorentzian space forms. After introducing the conformal compactification of the Lorentzian space forms, we define the conformal Gauss map which is a conformally invariant two parameter family of oriented spheres. We use the area of the conformal Gauss map to define the Willmore functional and derive a Bernstein type theorem for parabolic Willmore surfaces. Finally, we study the stability of maximal surfaces for the Willmore functional.Dedicated to Professor T.J. WillmoreSupported by an FPPI Postdoctoral Grant from DGICYT Ministerio de Educación y Ciencia, Spain 1994 and by a DGICYT Grant No. PB94-0750-C02-02     

Surfaces in Conformal Geometry

Properties of submanifolds are examined which remain invariantunder a conformal change of metric of the ambiant space. In particular,the Willmore energy functional is discussed as is the Willmoreconjecture for tori.     

Time-like Willmore surfaces in Lorentzian 3-space

Let R13 be the Lorentzian 3-space with inner product (, ). Let Q3 be the conformal compactification of R13, obtained by attaching a light-cone C∞ to R13 in infinity. Then Q3 has a standard conformal Lorentzian structure with the conformal transformation group O(3,2)/{±1}. In this paper, we study local conformal invariants of time-like surfaces in Q3 and dual theorem for Willmore surfaces in Q3. Let M (?) R13 be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p ∈ M we define S12(p) = {X ∈ R13 (X - c(P),X - c(p)) = 1/H(p)2} with c(p) = P 1/H(p)n(P) ∈ R13. Then S12 (p) is a one-sheet-hyperboloid in R3, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S12(p),p ∈ M} of hyperboloid in R13 defines in general two different enveloping surfaces, one is M itself, another is denoted by M (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in Q3 with non-degenerate associated surface M, then M is also a time-like Willmore surface in Q3 satisfying M = M; (ii) if M is a single point, then M is conformally equivalent to a minimal surface in R13.     

Surfaces with zero mean curvature vector in neutral 4-manifolds

Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and paraKähler surfaces, complex curves and paracomplex curves respectively are such surfaces and characterized by one additional condition. In neutral 4-dimensional space forms, the holomorphic quartic differentials defined on such surfaces vanish. There exist time-like surfaces with zero mean curvature vector and zero holomorphic quartic differential which are not compatible with the orientations of the spaces and the conformal Gauss maps of time-like surfaces of Willmore type and their analogues give such surfaces.     

Conformal CMC-Surfaces in Lorentzian Space Forms

Let Q3 be the common conformal compactification space of the Lorentzian space forms R13, S13 and H13. We study the conformal geometry of space-like surfaces in Q3. It is shown that any conformal CMC-surface in Q3 must be conformally equivalent to a constant mean curvature surface in R13, S13 or H13. We also show that if x : M→Q3 is a space-like Willmore surface whose conformal metric g has constant curvature K, then either K = - 1 and x is conformally equivalent to a minimal surface in R13, or K = 0 and x is conformally equivalent to the surface H1(1/(2~(1/2)))×H1(1/(2~(1/2))) in H13.     

Hamiltonian Stationary Lagrangian Surfaces in Euclidean Four-Space and Related Minimization Problems. Part II: Some Minimization Results

We study some minimization problems for Hamiltonian stationaryLagrangian surfaces in R⁴. We show that the flat Lagrangian torusS ¹ × S ¹ minimizes the Willmore functional among Hamiltonianstationary tori of its isotopy class, which gives a new proof of thefact that it is area minimizing in the same class. Considering theLagrangian flat cylinder as a torus in some quotient space R⁴/v Z, we show that it is also area minimizing in its isotopy class.     

Willmore Surfaces in S n

A surface x> : M S ⁿ is called a Willmore surface if it is a critical surface of the Willmore functional. It is well known that any minimal surface is a Willmore surface and that many nonminimal Willmore surfaces exists. In this paper, we establish an integral inequality for compact Willmore surfaces in S ⁿ and obtain a new characterization of the Veronese surface in S ⁴ as a Willmore surface. Our result reduces to a well-known result in the case of minimal surfaces.     

Willmore Surfaces of ?4 and the Whitney Sphere

We make a contribution to the study of Willmore surfaces infour-dimensional Euclidean space ⁴ by making useof the identification of ⁴ with two-dimensionalcomplex Euclidean space ². We prove that theWhitney sphere is the only Willmore Lagrangian surface of genus zero in⁴ and establish some existence and uniquenessresults about Willmore Lagrangian tori in ^{4 2}.     

Willmore surfaces of constant Möbius curvature

We study Willmore surfaces of constant Möbius curvature \({\mathcal{K}}\) in \({\mathbb{S}}^4\) . It is proved that such a surface in \({\mathbb{S}}^3\) must be part of a minimal surface in \({\mathbb{R}}^3\) or the Clifford torus. Another result in this paper is that an isotropic surface (hence also Willmore) in \({\mathbb{S}}^4\) of constant \({\mathcal{K}}\) could only be part of a complex curve in \({\mathbb{C}}^2 \cong {\mathbb{R}}^4\) or the Veronese 2-sphere in \({\mathbb{S}}^4\) . It is conjectured that they are the only possible examples. The main ingredients of the proofs are over-determined systems and isoparametric functions.     

Equivariant Willmore surfaces in conformal homogeneous three spaces

The complete classification of homogeneous three spaces is well known for some time. Of special interest are those with rigidity four which appear as Riemannian submersions with geodesic fibres over surfaces with constant curvature. Consequently their geometries are completely encoded in two values, the constant curvature, c$c$, of the base space and the so called bundle curvature, r$r$. In this paper, we obtain the complete classification of equivariant Willmore surfaces in homogeneous three spaces with rigidity four. All these surfaces appear by lifting elastic curves of the base space. Once more, the qualitative behaviour of these surfaces is encoded in the above mentioned parameters (c,r)$(c,r)$. The case where the fibres are compact is obtained as a special case of a more general result that works, via the principle of symmetric criticality, for bundle-like conformal structures in circle bundles. However, if the fibres are not compact, a different approach is necessary. We compute the differential equation satisfied by the equivariant Willmore surfaces in conformal homogeneous spaces with rigidity of order four and then we reduce directly the symmetry to obtain the Euler Lagrange equation of 4r²$4r^2$-elasticae in surfaces with constant curvature, c$c$. We also work out the solving natural equations and the closed curve problem for elasticae in surfaces with constant curvature. It allows us to give explicit parametrizations of Willmore surfaces and Willmore tori in those conformal homogeneous 3-spaces.     

Envelopes and osculates of Willmore surfaces

We view conformal surfaces in the 4-sphere as quaternionic holomorphiccurves in quaternionic projective space. By constructing envelopingand osculating curves, we obtain new holomorphic curves in quaternionicprojective space and thus new conformal surfaces. Applying theseconstructions to Willmore surfaces, we show that the osculatingand enveloping curves of Willmore spheres remain Willmore.     

Willmore energy estimates in conformal Berger spheres

We obtain isoperimetric inequalities for the Willmore energy of Hopf tori in a wide class of conformal structures on the three sphere. This class includes, on the one hand, the family of conformal Berger spheres and, on the other hand, a one parameter family of Lorentzian conformal structures. This allows us to give the best possible lower bound of Willmore energies concerning isoareal Hopf tori.     

Willmore子流形的共形高斯映射

本文研究球空间中子流形的共形高斯映射,用Moebius不变量刻划了该映射 为调和映射的条件.作为特例,指出球空间的2维子流形的共形高斯映射是调和映射 当且仅当该子流形是Willmore子流形.     

Minimizers of the Willmore functional with a small area constraint

We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we partially classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.     

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.     

Higher order Willmore hypersurfaces in Euclidean space

Let x ： Mn^n→ R^n＋1 be an n（≥2）-dimensional hypersurface immersed in Euclidean space Rn＋1. Let σi（0≤ i≤ n） be the ith mean curvature and Qn = ∑i=0^n（-1）^i＋1 （n^i）σ1^n-iσi. Recently, the author showed that Wn（x） = ∫M QndM is a conformal invariant under conformal group of R^n＋1 and called it the nth Willmore functional of x. An extremal hypersurface of conformal invariant functional Wn is called an nth order Willmore hypersurface. The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in Ra which have good geometric behaviors. The ordinary differential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper.     

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.