Let $\mathbf{C}^n \subset \mathbf{R}^{n+1}$ be an unstable minimal cone with an isolated singularity at the origin. Let $\Sigma \subset \partial B$ be its link, and $(\varphi_i)$ be the eigenfunctions of the Jacobi operator of $\Sigma$.
Let $M \subset B_1$ be a minimal surface in the unit ball, with boundary lying near $\Sigma$, but above it; specifically we take for some small $\delta > 0$, \begin{equation} \partial M = \operatorname{graph} \delta \varphi_1. \end{equation}
Is there something that prevents oscillations of $M$ in the 'axial' direction, meaning oscillations orthogonal to the 'radial' direction $\partial_r$?
To clarify the question, One way of putting the notion of 'oscillations' into words could be to take the cone over the boundary of $M$, that is $C_M = \{ t \omega \mid t \in (0,1), \omega \in \partial M \}$. (Naturally this is not minimal when $\delta > 0$.) One could then ask about the intersections of the slices $M_r$ with $C_M$. Say, how often does $M_r$ intersect $C_M$ when $r$ is not too small. Does $M_r$ lie on one side of $C_M$ when $r$ is close enough to one? If yes, can one estimate the first radius $r < 1$ for which $M_r \cap C_M \neq \emptyset$?
When the cone $\mathbf{C}$ is area-minimizing, then Brian White proved some estimates that can be used to control the 'axial' oscillation. I am thinking of Proposition 4.5 in the mapping degrees paper specifically. Basically what this states is that, as the higher modes of $\Sigma$ are dominated by $\varphi_1$ on the boundary, their respective contributions have to decrease (for the 'oscillating' higher modes) and increase (for the $\varphi_1$ mode). The situation is very different for unstable cones, mainly because $M$ could quickly diverge from $\mathbf{C}$, and so would fail to be graphical. From this point of view, I guess what I'm asking is whether some remnant of this remains true for unstable cones, too.
