Let $C$ be an algebraic curve of genus $g \geq 2$ defined over a number field $K$, and we suppose that $C(K) \ne \emptyset$ ($C$ may very well be defined over a proper subfield of $K$, but perhaps failing to have rational points over subfields). Then using the Abel-Jacobi map we can embed $C$ into its Jacobian $\text{Jac}(C)$, which is endowed with a Neron-Tate height $\hat{h}$. Applying this height to the image of $C$ in $\text{Jac}(C)$ we then obtain a height on $C$ itself.

Bogomolov's conjecture, proved by Emmanuel Ullmo and Shou-Wu Zhang in 1998, asserts that there exists $\epsilon > 0$ such that the set

$$\displaystyle \{x \in C(\overline{K}) : \hat{h}(x) < \epsilon \}$$

is finite. It thus follows that there exists an algebraic point on $C$ of *minimal* height.

My question is this: let $S(C)$ be the set of real numbers $r$ such that $\# \{x \in C(\overline{K}) : \hat{h}(x) \leq r\} = \infty$ and let $h_{\text{min}}(C) = \inf S(C)$. Can $h_{\text{min}}(C)$ be determined by other invariants of the curve, such as the Faltings height?