On null scrolls satisfying the condition \(\Delta H=AH\) (Q2706530)

Let \(\gamma\) be a null curve in a 3-dimensional Minkowski space \(E^3_1\) with the inner product \(g\), \(F=\{X,Y,Z\}\) a (proper) null frame field along the curve \(\gamma\), i.e. they satisfy NEWLINE\[NEWLINE\begin{aligned} g(X,X)= g(Y,Y)=0, \qquad & g(X,Y)=-1,\\ g(X,Z)=g(Y,Z)=0, \qquad & g(Z,Z)=1, \end{aligned}NEWLINE\]NEWLINE where \(X=k_0^{-1} \gamma'\), \(k_0\) is a given smooth positive function defined on \(\gamma\), \(Z=X \times Y\). The pair \((\gamma,F)\) is said to be a (proper) framed null curve. A nondegenerate ruled surface \(M\) along \(\gamma\) given by \(x(s,t)= \gamma(s)+t Y(s)\) is called a null scroll. A framed null curve \((\gamma,F)\) with \(k_0=1\), and \(X'(s)=k_2 (s)Z(s)\) is called a Cartan framed null curve, and the related null scroll is called a \(B\)-scroll. The main results are:NEWLINENEWLINENEWLINETheorem: Let \(M\) be a null scroll along the framed null curve with (proper) frame field. Then the mean curvature vector \(H\) satisfies \(\Delta H=AH\) for some \(A\in\text{Mat}(3,\mathbb{R})\) if and only if the mean curvature is constant. NEWLINENEWLINENEWLINE For related subjects see \textit{B.-Y. Chen} [Tamkang J. Math. 25, 71-81 (1994; Zbl 0811.53007)].