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Assuming the existence of a monster model, tameness and continuity ofnonsplitting in an abstract elementary class (AEC), we extend knownsuperstability results: let $\mu>LS({\bf K})$ be a regular stability cardinaland let $\chi$ be the local character of $\mu$-nonsplitting. The followingholds: 1. When $\mu$-nonforking is restricted to $(\mu,\geq\chi)$-limit modelsordered by universal extensions, it enjoys invariance, monotonicity,uniqueness, existence, extension and continuity. It also has local character$\chi$. This generalizes Vasey's result which assumed $\mu$-superstability toobtain same properties but with local character $\aleph_0$. 2. There is$\lambda\in[\mu,h(\mu))$ such that if ${\bf K}$ is stable in every cardinalbetween $\mu$ and $\lambda$, then ${\bf K}$ has $\mu$-symmetry while$\mu$-nonforking in (1) has symmetry. In this case (a) ${\bf K}$ has theuniqueness of $(\mu,\geq\chi)$-limit models: if $M_1,M_2$ are both$(\mu,\geq\chi)$-limit over some $M_0\in K_\mu$, then $M_1\cong_{M_0}M_2$; (b)any increasing chain of $\mu^+$-saturated models of length $\geq\chi$ has a$\mu^+$-saturated union. These generalize VanDieren-Vasey's result and removethe symmetry assumption in Boney-VanDieren and Vasey's result. Under$(<\mu)$-tameness, the conclusions of (1), (2)(a)(b) are equivalent to ${\bfK}$ having the $\chi$-local character of $\mu$-nonsplitting. Grossberg andVasey gave eventual superstability criteria for tame AECs with a monster model.We remove the high cardinal threshold and reduce the cardinal jump betweenequivalent superstability criteria. We also add two new superstability criteriato the list: a weaker version of solvability and the boundedness of the$U$-rank.

Stability results assuming tameness, monster model and continuity of nonsplitting