The maximal principle for properly immersed submanifolds and its applications

Luo, Yong

doi:10.1007/s10711-015-0114-4

Cited by 3 publications

(1 citation statement)

References 27 publications

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“…Moreover, in [15] Han introduced the notion of p-biharmonic submanifold and proved that p-biharmonic submanifold (M, g) in a Riemannian manifold (N, h) with non-positive sectional curvature which satis es certain condition must be minimal. Furthermore, Luo in [32,33] respectively generalized these results.…”

Section: Conjecture 12 Every Biharmonic Submanifold Of a Riemannianmentioning

confidence: 78%

F-biharmonic maps into general Riemannian manifolds

2019

Open Mathematics

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Let ψ:(M, g) → (N, h) be a map between Riemannian manifolds (M, g) and (N, h). We introduce the notion of the F-bienergy functional$$\begin{array}{} \displaystyle E_{F,2}(\psi)=\int\limits_{M}F\left(\frac{|\tau(\psi)|^{2}}{2}\right)\text{d}V_{g}, \end{array}$$where F : [0, ∞) → [0, ∞) be C3 function such that F′ > 0 on (0, ∞), τ(ψ) is the tension field of ψ. Critical points of τF,2 are called F-biharmonic maps. In this paper, we prove a nonexistence result for F-biharmonic maps from a complete non-compact Riemannian manifold of dimension m = dimM ≥ 3 with infinite volume that admit an Euclidean type Sobolev inequality into general Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the Lp-norm (p > 1) of the tension field is bounded and the m-energy of the maps is sufficiently small, then every F-biharmonic map must be harmonic. We also get a Liouville-type result under proper integral conditions which generalize the result of [Branding V., Luo Y., A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds, 2018, arXiv: 1806.11441v2].

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Section: Conjecture 12 Every Biharmonic Submanifold Of a Riemannianmentioning

confidence: 78%

F-biharmonic maps into general Riemannian manifolds

2019

Open Mathematics

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Nonexistence of proper p-biharmonic maps and Liouville type theorems I: case of $$p\ge 2$$

Han

Luo

2019

J Elliptic Parabol Equ

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Let u : (M, g) → (N, h) be a map between Riemannian manifolds (M, g) and (N, h). The p-bienergy of u is defined by E p (u) = M |τ (u)| p dν g , where τ (u) is the tension field of u and p > 1. Critical points of E p (·) are called p-biharmonic maps. In this paper we will prove nonexistence result of proper p-biharmonic maps when p ≥ 2. In particular when M = R m , we get Liouville type results under proper integral conditions , which extend the related results of Baird, Fardoun and Ouakkas [1].

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Biharmonic hypersurfaces in a sphere

Luo

Maeta

2017

Proc. Amer. Math. Soc.

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In this short paper we will survey some recent developments in the geometric theory of biharmonic submanifolds, with an emphasis on the newly discovered Liouville type theorems and applications of known Liouville type theorems in the research of the nonexistence of biharmonic submanifolds. A new Liouville type theorem for superharmonic functions on complete manifolds is proved and its applications in a kind of nonexistence of biharmonic hypersurfaces in a sphere is provided.

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The maximal principle for properly immersed submanifolds and its applications

Cited by 3 publications

References 27 publications

F-biharmonic maps into general Riemannian manifolds

F-biharmonic maps into general Riemannian manifolds

Nonexistence of proper p-biharmonic maps and Liouville type theorems I: case of $$p\ge 2$$

Biharmonic hypersurfaces in a sphere

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