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In this article, we focus on $L^{2}(\mathbb{R}^d)\times\cdots\timesL^{2}(\mathbb{R}^d)\rightarrow L^{2/m}(\mathbb{R}^d)$ estimates for multilinearmaximal averages over non-degenerate hypersurfaces. Our findings is new for$m$-linear averages with $m\geq3$, and represent a reproof of the recent resultof T. Borges, B. Foster, and Y. Ou on the curvature conditions of thehypersurfaces required in establishing $L^{2}(\mathbb{R}^d)\timesL^{2}(\mathbb{R}^d)\rightarrow L^{1}(\mathbb{R}^d)$ estimates of bilinearmaximal functions.

Improved curvature conditions on $L^2\times\cdots\times L^2 \to L^{2/m}$ bounds for multilinear maximal averages