3-manifolds admitting locally large distance 2 Heegaard splittings

It is known that every closed, orientable 3-manifold admits a Heegaard splitting. By Thurston’s Geometrization conjecture, proved by Perelman, a 3-manifold admitting a Heegaard splitting of distance at least 3 is hyperbolic. So what about 3-manifolds admitting distance at most 2 Heegaard splittings? Inspired by the construction of hyperbolic 3-manifolds in [Qiu, Zou and Guo, Pacific J. Math. 275 (2015), no. 1, 231-255], we introduce the definition of a locally large geodesic in curve complex and also a locally large distance 2 Heegaard splitting. Then we prove that if a 3-manifold admits a locally large distance 2 Heegaard splitting, then it is either a hyperbolic 3-manifold or an amalgamation of a hyperbolic 3-manifold and a small Seifert fiber space along an incompressible torus. After examining those non hyperbolic cases, we give a sufficient and necessary condition to determine a hyperbolic 3-manifold admitting a locally large distance 2 Heegaard splitting.