Singular Lagrangians in the Hitchin moduli space and conformal limits

In the moduli space of semistable $\text{SL}(r,\mathbb{C})$-Higgs bundles, we show that there exists a sublocus of the upward flow through a polystable ${\mathbb{C}}^{\ast }$-fixed point, which is Lagrangian on its intersection with the stable locus. This intesesction is always non-empty in the case when the Higgs field of the fixed point vanishes, or when the automorphism group of its polystable representative is abelian. Under the same assumptions, we show that the conformal limit of a stable Higgs bundle lying on this locus exists.