Are there hyperkahler manifolds with multi-isotropic hyperkahler submanifolds?

Question

This is a continuation of this question. As explained in the comments, for a hyperkaehler manifold $X$, a multi-isotropic submanifold $S$ has the property that the three distinguished Kahler forms $\omega^u$ of $X$ vanish when pulled back to $S$, i.e., $$\omega^u(T_1,T_2)=0$$ for tangent vectors $T_1,T_2$ of $S$.

My question is, can such a multi-isotropic submanifold $S$ be found such that $S$ itself is hyperkaehler?

The maximum dimension for $S$ (when $X$ is 4k-dimensional) is $k$. Perhaps it could be shown that for a hyperkaehler manifold $\tilde{S}$, the space $T^*\tilde{S}\times T^*\tilde{S}$ is hyperkaehler, for which $\tilde{S}$ is a multi-isotropic submanifold?

Answer 1

NB: New evidence has changed my conclusions.

In the first nontrivial case where this question makes sense, i.e., when $X$ has dimension $16$ and $S\subset X$ has dimension $4$, a preliminary calculation suggested that the answer might be 'yes', but now, further analysis casts doubt on this conclusion.

Of course, there are examples in which this happens: For example, if we give $\mathbb{R}^{16}$ the flat hyperKähler structure, then it contains multi-isotropic $4$-planes that are flat and hence the induced metric has holonomy in $\mathrm{SU}(2)\subset\mathrm{SO}(4)$ and hence have compatible hyperKähler structures. The real question is whether such multi-isotropic submanifolds with induced hyperKähler metrics exist in the general hyperKähler $16$ manifold and, if so, how plentiful they are, and that is a much harder question.

First, one might want to know how plentiful are the multi-isotropic $4$-manifolds. This can be answered. The condition for $S^4\subset X^{16}$ to be multi-isotropic is an overdetermined system of $18$ first-order PDE for $S^4$, and calculation shows that it is involutive, with last nonzero Cartan character $s_4 = 3$, so that the 'generic' such multi-isotropic $4$-dimensional submanifold depends on $3$ functions of four variables, locally.

Changed paragraph: Now, if one adds the condition that the induced metric on $S^4$ have holonomy in $\mathrm{SU}(2)\subset\mathrm{SO}(4)$ (which is the hyperKähler condition in dimension $4$), the larger system (which includes second order equations as well as first order equations), is not involutive, unfortunately. (I originally thought that it might be, but a careful calculation today turned up an error in my earlier calculations.) Since the PDE system is not involutive, it needs to be prolonged to check for existence (and generality) of solutions, and, to do this, the curvature of the ambient hyperKähler metric in dimension $16$ will have to be taken into account, greatly complicating the problem. I now suspect (though I don't know for sure) that the answer would eventually come out that, for the generic hyperKähler metric in dimension $16$, none of the multi-isotropic $4$ manifolds will inherit an induced metric with holonomy in $\mathrm{SU}(2)$ (which equivalent to it being hyperKähler).