L<sub>K</sub>-BIHARMONIC HYPERSURFACES IN SPACE FORMS WITH THREE DISTINCT PRINCIPAL CURVATURES
L<sub>K</sub>-BIHARMONIC HYPERSURFACES IN SPACE FORMS WITH THREE DISTINCT PRINCIPAL CURVATURES
- 대한수학회
- Communications of the Korean Mathematical Society
- Vol.35 No.4
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2020.011221 - 1244 (24 pages)
- 0
In this paper we consider L<sub>K</sub>-conjecture introduced in [5, 6] for hypersurface M<sup>n</sup> in space form R<sup>n+1</sup>(c) with three principal curvatures. When c = 0, -1, we show that every L<sub>1</sub>-biharmonic hypersurface with three principal curvatures and H<sub>1</sub> is constant, has H<sub>2</sub> = 0 and at least one of the multiplicities of principal curvatures is one, where H<sub>1</sub> and H<sub>2</sub> are first and second mean curvature of M and we show that there is not L<sub>2</sub>-biharmonic hypersurface with three disjoint principal curvatures and, H<sub>1</sub> and H<sub>2</sub> is constant. For c = 1, by considering having three principal curvatures, we classify L<sub>1</sub>-biharmonic hypersurfaces with multiplicities greater than one, H<sub>1</sub> is constant and H<sub>2</sub> = 0, proper L<sub>1</sub>-biharmonic hypersurfaces which H<sub>1</sub> is constant, and L<sub>2</sub>-biharmonic hypersurfaces which H<sub>1</sub> and H<sub>2</sub> is constant.
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