In this paper, we give an improved Morse index bound of minimal hypersurfaces from Almgren-Pitts min-max construction in any closed Riemannian manifold $M^{ n+1}$ $(n+1 \geq 3$), which generalizes a result by X. Zhou \cite{ zhou_multiplicity_2019} for $3 \leq n+1 \leq 7$. The novel techniques are the construction of hierarchical deformations and the restrictive min-max theory. These techniques do not rely on bumpy metrics, and thus could be adapted to many other min-max settings.
Categories: math.DG math.AP