We show that subsurfaces of a Heegaard surface for which the relative Hempel distance of the splitting is sufficiently high have to appear in any Heegaard surface of genus bounded by half that distance.
Jesse Johnson. Yair Minsky. Yoav Moriah. "Heegaard splittings with large subsurface distances." Algebr. Geom. Topol. 10 (4) 2251 - 2275, 2010. https://doi.org/10.2140/agt.2010.10.2251