This is related to this other question question of mine.
Let $M$ be a $3$-dimensional closed (compact without boundary) strongly pseudoconvex manifold and let $\Delta_t$ be a collection of differential operators with $t$ in a small $2$-dimensional real disk, that is, $t \in D_\epsilon^2$ . We look for existence (or obstruction to existence) of solutions $f_t:M \to \mathbb{R}^2$ satisfying the equation $$\Delta_t f_t = 0$$ that extend an initial solution $\Delta_0 f_0 = 0$ to a small neighborhood of $0 \in D^2_\epsilon$. I just want $f_t$ to depend continuously on $t$ and you can assume that the dependence of $\Delta_t$ on $t$ is analytic.
My question is
What results are in the literature that impose restrictions on $\Delta_t$ or the topology of $M$ so that solutions exist for a small disk inside $D_\epsilon^2$?
This questions seems too strong and that the answer is in general negative. So we can relax to
What if we attempt to find solutions only on a small curve (or union of curves) passing through $0 \in D^2_\epsilon$?
This is much weaker and seems possible that some existence result might be already on the literature.
