Deformations of Legendre curves. (English) Zbl 0907.53037

The space \({\mathbb R}^3\) with the standard Darboux form \(\eta= {1\over 2}(dz-ydx)\) and Sasakian metric \(g={1\over 4}(dx^2+dy^2)+\eta \otimes \eta\) is the Sasakian manifold \({\mathbb R}^3(-3)\) and Legendre curves in \({\mathbb R}^3(-3)\) are the 1-dimensional integral submanifolds, namely the curves \(\gamma (t)= (x(t), y(t), z(t))\in {\mathbb R}^3(-3)\) for which \(\eta (\gamma')=0\).
In this paper, the authors study deformations of Legendre curves in \({\mathbb R}^3(-3)\) in the direction of the principal normal, especially 2-minimal curves. First, the authors recall from [B.-Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Kyushu J. Math. 49, 103-121 (1995; Zbl 0862.53043)] the notions of \(k\)-deformations and \(k\)-minimality as follows.
For an oriented compact hypersurface \(x: M^n\to \widetilde M^{n+1} \) with normal vector field \(\xi \), let \(\varphi _t : M^n\to \widetilde M^{n+1} \) be a family of normal deformations given by \(\varphi _t(p)= \exp_{x(p)}tf(p)\xi(p), p\in M, t\in (-\varepsilon, \varepsilon),\) for sufficiently small \(\varepsilon\). Then for each \(k\in {\mathbb N},\) define \( {\mathfrak {F}}_k\) to be the class of normal deformations associated to functions \(f\) which are \(L^2\)-orthogonal to the eigenspaces \(V_q\) of the Laplacian for \(q<k\). If \(A(t)\) denotes the volume of \(M_t=\varphi _t(M),\) then \(M\) is said to be \(k\)-minimal if \(A'(0)=0\) for all deformations on \( {\mathfrak {F}}_k\).
Here the authors are concerned with closed Legendre curves \(\gamma : [0, L] \to {\mathbb R}^3(-3)\) parametrized by arc length and they study the length integral \(L(t)\) under deformations in the direction of the principal normal. A closed curve \(\gamma \) is \(k\)-minimal if \(L'(0)=0\) for all deformations in \( {\mathfrak {F}}_k\).
The main results of the present paper are the following. (1) Every closed 2-minimal curve in the plane gives rise to a closed 2-minimal Legendre curve \(\gamma \) in \({\mathbb R}^3(-3)\) by integration of \(z'=yx'\) and conversely. (2) Every closed 2-minimal curve in \(E^2\) has a point of self-intersection and algebraic area zero.
Also, in the last section the authors consider deformations of curves in a general \(K\)-contact manifold in the direction of the characteristic vector field and prove that critical curves for this variational problem are the \(C\)-loxodromes.


53C40 Global submanifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53A04 Curves in Euclidean and related spaces


Zbl 0862.53043