• 대한수학회지
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• 제54권4호
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• pp.1189-1208
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• 2017

In this paper, we study vanishing properties for $L^2$ harmonic 1-forms on a gradient shrinking Ricci soliton. We prove that if (M, g, f) is a complete oriented noncompact gradient shrinking Ricci soliton with potential function f, then there are no non-trivial $L^2$ harmonic 1-forms which are orthogonal to df. Second, we show that if the scalar curvature of the metric g is greater than or equal to (n - 2)/2, then there are no non-trivial $L^2$ harmonic 1-forms on (M, g). We also show that any multiplication of the total differential df by a function cannot be an $L^2$ harmonic 1-form unless it is trivial. Finally, we derive various integral properties involving the potential function f and $L^2$ harmonic 1-forms, and handle their applications.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제30권2호
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• pp.109-129
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• 2023

In this paper, we introduce a second order linear differential operator ^T□: C^∞ (M) → C^∞ (M) as a natural generalization of Cheng-Yau operator, [8], 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 ^T□ 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 fT-harmonic maps for conformal immersions and as application of it, we consider fL_k-harmonic hypersurfaces in space forms, and after that we classify complete fL₁-harmonic surfaces, some fL_k-harmonic isoparametric hypersurfaces, fL_k-harmonic weakly convex hypersurfaces, and we show that there exists no compact fL_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.

• 대한수학회보
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• 제60권1호
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• pp.171-184
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• 2023

In this paper, we show that a complete translating soliton Σ^m in ℝⁿ for the mean curvature flow is stable with respect to weighted volume functional if Σ satisfies that the L^m norm of the second fundamental form is smaller than an explicit constant that depends only on the dimension of Σ and the Sobolev constant provided in Michael and Simon [12]. Under the same assumption, we also prove that under this upper bound, there is no non-trivial f-harmonic 1-form of L²_f on Σ. With the additional assumption that Σ is contained in an upper half-space with respect to the translating direction then it has only one end.