Let $X$ be an algebraic curve over $\mathbb C$ (feel free to add smoothness or other usefule properties). Fix degrees $d_1,d_2$ and ranks $r_1,r_2$. The slopes are $\mu_i=d_i/r_i$. Assume $\mu_1>\mu_2$.
Let $E_i$ be a semistable vector bundle of degree $d_i$ and rank $r_i$ for $i=1,2$. Semistability of a vector bundle $E$ is defined as the condition that $\mu(F):=\frac{\deg F}{\mathrm{rank}F}\leq\frac{\deg E}{\mathrm{rank}E}$ for any subbundle $F\subset E$.
I know little about this topic, and now I want to know about the space $$\mathrm{Hom}(E_2,E_1).$$
The following question is what I am interested:
What is the minimum of the dimension of $\mathrm{Hom}(E_2,E_1)$, when $E_1,E_2$ range over semistable bundles of prescribed degrees and ranks?
The motivation comes from considering automorphism groups of vector bundles of fixed Harder-Narasimhan type of length 2. If $0\subsetneq E_1\subsetneq E$ is the HN filtration, then the Lie algebra of the unipotent stabiliser is $\mathrm{Hom}(E/E_1,E_1)$.