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.     

New Examples of Willmore Surfaces in S n

A surface x: M S ⁿ is called a Willmore surface if it is a criticalsurface of the Willmore functional _M (S – 2H ²)dv, where H isthe mean curvature and S is the square of the length of the secondfundamental form. It is well known that any minimal surface is aWillmore surface. The first nonminimal example of a flat Willmoresurface in higher codimension was obtained by Ejiri. This example whichcan be viewed as a tensor product immersion of S ¹(1) and a particularsmall circle in S ²(1), and therefore is contained in S ⁵(1) gives anegative answer to a question by Weiner. In this paper we generalize theabove mentioned example by investigating Willmore surfaces in S ⁿ (1)which can be obtained as a tensor product immersion of two curves. We inparticular show that in this case too, one of the curves has to beS ¹(1), whereas the other one is contained either in S ²(1) or in S ³(1). In the first case, we explicitly determine the immersion interms of elliptic functions, thus constructing infinetely many newnonminimal flat Willmore surfaces in S ⁵. Also in the latter casewe explicitly include examples.     

Willmore surfaces in the four-sphere

Willmore immersions of an orientable surface X in the n-dimensionalsphere appear as the extremal points of a conformally invariant variational problem in the space of all immersions f: X S ⁿ.In this paper we will study Willmore immersions of the differentiable two-sphere in S ⁴, using the method of moving frames and Cartan's conformal structures.The work on this paper was partially supported by a Fellowship of the Consiglio Nazionale delle Ricerche.     

Willmore Lagrangian Spheres in the Complex Euclidean Space C n

In this paper we construct many examples of n-dimensionalWillmore Lagrangian submanifolds in the complex Euclidean space C ⁿ . We characterize them as the only Willmore Lagrangian submanifolds invariant under the action of SO(n). The mostimportant contribution of our construction is that it provides examplesof Willmore Lagrangian spheres in C ⁿ for all n 2.     

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.     

Analysis of constrained Willmore surfaces

This paper investigates the regularity of constrained Willmore immersions into ?^m≥3 locally around both “regular” points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions for the immersion and its first and second derivatives, given in terms of residues computed as circulation integrals. We deduce explicit “point removability” conditions ensuring that the immersion is smooth. Our results apply in particular to Willmore immersions and to parallel mean curvature immersions in any codimension.     

A new characterization of Willmore submanifolds

Let M ⁿ be a compact Willmore submanifold in the unit sphere S ^n+p . In this note, we investigate the first eigenvalue of the Schrödinger operator L = ?Δ?q on M, where q is some potential function on M, and present a gap estimate for the first eigenvalue of L.     

On One Family of Minimal Tori in ℝ<Superscript>3</Superscript> with Planar Embedded Ends

We construct new examples of complete minimal tori in the three-dimensional Euclidean space with an arbitrary even number n ≥ 6 of planar embedded ends.     

There exist no 2-type surfaces in E 3 which are images under stereographic projection of minimal surfaces in S 3

In this paper we deal with the following particular case of a weaker conjecture by B. Y. Chen: Are there 2-type Willmore surfaces in E ³? In particular we prove that the above question has a negative answer when the surface is the image under stereographic projection of a minimal surface in S ³.     

Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional

We study the generalization of the Willmore functional for surfaces in the three-dimensional Heisenberg group. Its construction is based on the spectral theory of the Dirac operator entering into theWeierstrass representation of surfaces in this group. Using the surfaces of revolution we demonstrate that the generalization resembles the Willmore functional for the surfaces in the Euclidean space in many geometrical aspects. We also observe the relation of these functionals to the isoperimetric problem.     

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S4, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them the interesting duality theorem holds. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.     

On spectral characterizations of Willmore hypersurfaces in a sphere

Let M be a closed Willmore hypersurface in the sphere S^n＋1（1） （n ≥ 2） with the same mean curvature of the Willmore torus Wm,n-m, if SpecP（M） = Spec^P（Wm,n-m ） （p = 0, 1,2）, then M is Wm,n-m.     

Willmore submanifolds in the unit sphere via isoparametric functions

In this paper, we show that both focal submanifolds of each isoparametric hypersurface in the sphere with six distinct principal curvatures are Willmore, hence all focal submanifolds of isoparametric hypersurfaces in the sphere are Willmore.     

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.     

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.     

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.     

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.     

The Willmore functional of surfaces

The Whitney immersion theorem asserts that every smooth n-dimensional manifold can be immersed in R2n-1, in particular, smooth surfaces can be immersed in R3, and for embeddings the ambient dimension needs to go up by 1. Immersions or embeddings carry both intrinsic and extrinsic information of manifolds, and the latter determines how the manifold fits into the Euclidean space. The key geometric quantity to capture the extrinsic geometry is the second fundamental form.