Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic
in Lorentzian conformal geometry which parallels the theory of Willmore
surfaces in $S^4$, 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 holds interesting duality theorem. As an application spacelike
Willmore 2-spheres are classified. Finally we construct a family of homogeneous
spacelike Willmore tori.Comment: 19 page
In this paper we provide a systematic discussion of how to incorporate orientation preserving symmetries into the treatment of Willmore surfaces via the loop group method.In this context we first develop a general treatment of Willmore surfaces admitting orientation preserving symmetries, and then show how to induce finite order rotational symmetries. We also prove, for the symmetric space which is the target space of the conformal Gauss map of Willmore surfaces in spheres, the longstanding conjecture of the existence of meromorphic invariant potentials for the conformal Gauss maps of all compact Willmore surfaces in spheres. We also illustrate our results by some concrete examples. *
The family of Willmore immersions from a Riemann surface into S n+2 can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in R n+2 and those which are not conformally equivalent to a minimal surface in R n+2 . On the level of their conformal Gauss maps into Gr 1,3 (R 1,n+3 ) = SO + (1, n + 3)/SO + (1, 3) × SO(n) these two classes of Willmore immersions into S n+2 correspond to conformally harmonic maps for which every image point, considered as a 4-dimensional Lorentzian subspace of R 1,n+3 , contains a fixed lightlike vector or where it does not contain such a "constant lightlike vector". Using the loop group formalism for the construction of Willmore immersions we characterize in this paper precisely those normalized potentials which correspond to conformally harmonic maps containing a lightlike vector. Since the special form of these potentials can easily be avoided, we also precisely characterize those potentials which produce Willmore immersions into S n+2 which are not conformal to a minimal surface in R n+2 . It turns out that our proof also works analogously for minimal immersions into the other space forms.
The conformal geometry of spacelike surfaces in 4-dim Lorentzian space forms has been studied by the authors in a previous paper, where the so-called polar transform was introduced. Here it is shown that this transform preserves spacelike conformal isothermic surfaces. We relate this new transform with the known transforms (Darboux transform and spectral transform) of isothermic surfaces by establishing the permutability theorems.
Mathematics Subject Classification (2000). Primary 53B25; Secondary 53B30.
The paper builds a DPW approach of Willmore surfaces via conformal Gauss maps. As applications, we provide descriptions of minimal surfaces in R n+2 , isotropic surfaces in S 4 and homogeneous Willmore tori via the loop group method. A new example of a Willmore two-sphere in S 6 without dual surfaces is also presented.
We solve the analogue of Björling's problem for Willmore surfaces via a harmonic map representation. For the umbilic-free case the problem and solution are as follows: given a real analytic curve y 0 in S 3 , together with the prescription of the values of the surface normal and the dual Willmore surface along the curve, lifted to the light cone in Minkowski 5-space R 5 1 , we prove, using isotropic harmonic maps, that there exists a unique pair of dual Willmore surfaces y andŷ satisfying the given values along the curve. We give explicit formulae for the generalized Weierstrass data for the surface pair. For the three dimensional target, we use the solution to explicitly describe the Weierstrass data, in terms of geometric quantities, for all equivariant Willmore surfaces. For the case that the surface has umbilic points, we apply the more general halfisotropic harmonic maps introduced by Hélein to derive a solution: in this case the mapŷ is not necessarily the dual surface, and the additional data of a derivative ofŷ must be prescribed. This solution is generalized to higher codimensions.
Applying the general theory about complete spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space R 4 1 , we classify those regular algebraic ones with total Gaussian curvature − KdM = 4π. Such surfaces must be oriented and be congruent to either the generalized catenoids or the generalized enneper surfaces. For non-orientable stationary surfaces, we consider the Weierstrass representation on the oriented double covering M (of genus g) and generalize Meeks and Oliveira's Möbius bands. The total Gaussian curvature are shown to be at least 2π(g + 3) when M → R 4 1 is algebraic-type. We conjecture that there do not exist non-algebraic examples with − KdM = 4π.
Conserved quantities of the Cosserat elastic rod dynamics are studied according to the general theorems of dynamics. The rod dynamical equation takes the cross section of the rod as its objective of study and is expressed by two independent variables, the arc coordinate of the rod and the time, so the conserved quantities are written in the integral forms and there exist the arc coordinate conservation and the time conservation. The existence conditions and the formulas of conservations of momentum and moment of momentum are derived from the theorem of momentum and the theorem of moment of momentum respectively, which contain two cases of conserved quanties, one is the time and the other is arc coordinate. Also existence conditions and formulas of conservations of energy about time and are coordinate, which contain mechanical energy conservation, are derived from energy equations about the time and arc coordinate of the rod respectively. All of conservative motions of the rod are explained by examples. The conserved quantities in the integral form are of practical significance in both theoretical and numerical analysis for the Cosserat elastic rod dynamics.
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