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- 저자 : Mehran Aminian (검색결과 3 건)
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{$L_k$}-biharmonic hypersurfaces in space forms with three distinct principal curvatures
Mehran Aminian 대한수학회 2020 대한수학회논문집 Vol.35 No.4
In this paper we consider $ L_k $-conjecture introduced in \cite{AminianKashani, AminKashani} 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.
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L<sub>K</sub>-BIHARMONIC HYPERSURFACES IN SPACE FORMS WITH THREE DISTINCT PRINCIPAL CURVATURES
Aminian, Mehran Korean Mathematical Society 2020 대한수학회논문집 Vol.35 No.4
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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Introduction of T-harmonic Maps
Mehran Aminian 한국수학교육학회 2023 純粹 및 應用數學 Vol.30 No.2
In this paper, we introduce a second order linear differential operator $ \stackrel{T}{\Box}:C^\infty(M)\rightarrow C^\infty(M) $ as a natural generalization of Cheng-Yau operator, \cite{Yauoperator}, where $ T $ is a $ (1,1) $-tensor on Riemannian manifold $ (M,h) $, and then we show on compact Riemannian manifolds, divT=divT^t, and if divT = 0, and f be a smooth function on M, the condition $ \stackrel{T}{\Box}f=0 implies that f is constant. Hereafter, we introduce T-energy functionals and by deriving variations of these functionals, we define T-harmonic maps between Riemannian manifolds, which is a generalization of L_{k}-harmonic maps introduced in [3]. Also we have studied f$ T $-harmonic maps for conformal immersions and as application of it, we consider f$ L $ _{k}-harmonic hyper- surfaces in space forms, and after that we classify complete f$L$_{1}-harmonic surfaces, some f$L$_{k}-harmonic isoparametric hypersurfaces, f$L$_{k}-harmonic weakly convex hypersurfaces, and we show that there exists no compact f$L$_{k}-harmonic hypersurface either in the Euclidean space or in the hyperbolic space or in the Euclidean hemisphere. As well, some properties and examples of these definitions are given.
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