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 $L_k$-biharmonic hypersurfaces in space forms with three distinct principal curvatures Commun. Korean Math. Soc.Published online August 5, 2020 Mehran Aminian Faculty of Dept. of Math., Vali-e-Asr University of Rafsanjan Abstract : In this paper we consider $L_k$-conjecture introduced in [5,6] for hypersurface $M^n$ in space form $R^{n+1}(c)$ with three principal curvatures. When $c=0, -1$, we show that every $L_1$-biharmonic hypersurface with three principal curvatures and $H_1$ is constant, has $H_2=0$ and at least one of the multiplicities of principal curvatures is one, where $H_1$ and $H_2$ are first and second mean curvature of $M$ and we show that there is not $L_2$-biharmonic hypersurface with three disjoint principal curvatures and, $H_1$ and $H_2$ is constant. For $c=1$, by considering having three principal curvatures, we classify $L_1$-biharmonic hypersurfaces with multiplicities greater than one, $H_1$ is constant and $H_2=0$, proper $L_1$-biharmonic hypersurfaces which $H_1$ is constant, and $L_2$-biharmonic hypersurfaces which $H_1$ and $H_2$ is constant. Keywords : $L_k$ operator; biharmonic hypersurfaces; $L_k$-conjecture MSC numbers : 53C40; 53C42 Full-Text :