Min-max minimal hypersurfaces with higher multiplicity

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We exhibit the first set of examples of non-bumpy metrics on the ( $n+1$ )-sphere $(2 \leq n \leq 6)$ in which the varifold associated with the two-parameter min-max construction must be a multiplicity-two minimal $n$ -sphere. This is proved by a new area-and-separation estimate for certain minimal hypersurfaces with Morse index two inspired by an early work of Colding-Minicozzi. We also construct non-bumpy projective spaces in which the first min-max hypersurfaces are one-sided, and non-bumpy balls in which the free boundary min-max hypersurfaces are improper.