It's well known that maps $\mathbb{P}^1_\mathbb{C} \to A$ are constant for any Abelian variety $A$ (in fact, for any complex torus).

Is there any similar statement in the tropical case? Naively, the answer seems to be no. We can see this in the following way: a tropical $\mathbb{P}^1$, $\mathbb{TP}^1$ is just a metric tree, and (for example) a circle (with a view vertices for good measure) produces an example of a one-dimensional tropical abelian variety. We can include $\mathbb{TP}^1 \hookrightarrow C$ as one of the edges quite easily which is a tropical map (unless I'm making a silly mistake).

So naively, the answer seems to be that there *are* non-constant tropical maps. Is there some way to eliminate these in a meaningful way? That is, it would be nice to have a statement of the form

**Theorem**: All (possibly restricted) tropical maps $\mathbb{TP}^1 \to A$ are constant (up to some appropriate notion of equivalence).

**Edit:** As pointed out in the comments, there is a balancing condition to be considered. For the case of maps to a tropical elliptic curve, this is given as the following.

For each point in $C$, the target, and each pre-image $p$ of $C$, the sum of all weights of the edges incident to $p$ must sum to zero (paying attention to outward orientation of edges).

One sees immediately that this implies that no $\mathbb{TP}^1$s can map to $C$, since the image of a leaf of a tree can never satisfy the balancing condition (unless the weight of every edge is zero).

However, this doesn't address maps to higher dimensional abelian varieties, but one can hope that a similar balancing condition would be true.