New Examples of Willmore Tori in S4
نویسندگان
چکیده
Using the (generalized) Darboux transformation in the case of the Clifford torus, we construct for all Pythagorean triples (p, q, n) ∈ Z a CP–family of Willmore tori in S with Willmore energy 2nπ.
منابع مشابه
Generalized Weierstrass Formulae, Soliton Equations and Willmore Surfaces I. Tori of Revolution and the Mkdv Equation
A new approach is proposed for study structure and properties of the total squared mean curvature W of surfaces in R 3. It is based on the generalized Weierstrass formulae for inducing surfaces. The quantity W (Will-more functional or extrinsic Polyakov action) is shown to be invariant under the modified Novikov–Veselov hierarchy of integrable flows. It is shown that extremals of W (Willmore su...
متن کاملHopf Tori in S 3
Garsia [23 had shown that every compact Riemann surface (of any genus) can be conformally embedded in ~3 as an algebraic surface, but his method of proof was not constructive and he therefore did not obtain bounds for the degree of this surface. As another application of Hopf tori we construct new examples of compact embedded Willmore surfaces. A surface in ~3 is called a Willmore surface if it...
متن کاملWillmore-chen Tubes on Homogeneous Spaces in Warped Product Spaces
We present a new method to obtain Willmore-Chen submanifolds in spaces endowed with warped product metrics and fibers being a given homogeneous space. The main points are: First the invariance of the variational problem of WillmoreChen with respect to the conformal changes in the ambient space metric. Second, the principle of symmetric criticality which allows us to relate the problem with that...
متن کاملThe Willmore Functional on Lagrangian Tori: Its Relation to Area and Existence of Smooth Minimizers
In this paper we prove an existence and regularity theorem for la-grangian tori minimizing the Willmore functional in Euclidean four-space, R4 ,with the standard metric and symplectic structure. Technical difficulties arisebecause the Euler-Lagrange equation for this problem is a sixth-order nonlinearpartial differential equation. This research was motivated by a study of the<lb...
متن کاملWillmore Submanifolds in a Sphere
Let x : M → Sn+p be an n-dimensional submanifold in an (n + p)dimensional unit sphere Sn+p, x : M → Sn+p is called a Willmore submanifold if it is an extremal submanifold to the following Willmore functional: ∫ M (S − nH) 2 dv, where S = ∑ α,i,j (hij) 2 is the square of the length of the second fundamental form, H is the mean curvature of M . In [13], author proved an integral inequality of Sim...
متن کامل