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.

MSC:

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

Zbl 0862.53043