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.

Weierstrass Type Representation of Willmore Surfaces in S^n

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.

Conformal CMC-Surfaces in Lorentzian Space Forms

Let Q^3 be the common conformal compactification space of the Lorentzian space forms Q^3 1 ,S^3 1,We study the conformal geometry of space-like surfaces in Q^3 ,It is shown that any conformal CMC-surface in Q^3 must be conformally equivalent to a constant mean curvature surface in R^3 1,or,H^3 1,We also show that if x ：M→Q^3 is a space-like Willmore surface whose conformal metric g has constant curvature K,the either K = -1 and x is conformally equivalent to a minimal surface in R^3 1,or K=0 and x is conformally equivalent to the surface H^1（1/√2）×H^1（1/√2） in H^3 1.

Existence and uniqueness for variational problem from progressive lens design

We study a functional modelling the progressive lens design,which is a combination of Willmore functional and total Gauss curvature.First,we prove the existence for the minimizers of this class of functionals among the class of revolution surfaces rotated by the curves y=f(x)about the x-axis.Then,choosing such a minimiser as background surfaces to approximate the functional by a quadratic functional,we prove the existence and uniqueness of the solution to the Euler-Lagrange equation for the quadratic functionals.Our results not only provide a strictly mathematical proof for numerical methods,but also give a more reasonable and more extensive choice for the background surfaces.

A Characterization of the Ejiri Torus in S^5

We conjecture that a Willmore torus having Willmore functional between 2π2 and 2π2 √3 is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri＇s torus in S5 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 S3, 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 S5 attains the minimum 2π2 √3, which indicates our conjecture holds true for Wilhnore 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 S5. Moreover, similar to Li and Vrancken, we classify all constrained Wilhnore surfaces of tensor product by reducing them with elastic curves in S3. All constrained Willmore tori obtained this way are also shown to bc unstable when the co-dimension is big enough.