Rigidity of self-shrinkers and translating solitons of mean curvature flows

Chen, Qun; Qiu, Hongbing

doi:10.1016/j.aim.2016.03.004

Cited by 61 publications

(33 citation statements)

References 26 publications

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“…Firstly, we classify m-dimensional complete spacelike translating solitons in R m+n n by affine technique and classical gradient estimates, and prove the only complete spacelike translating solitons in R m+n n are the spacelike m-planes. This result provides another proof of a nonexistence theorem for complete spacelike translating solitons in [8], and a simple proof of rigidity theorem in [33]. Secondly, we generalize the rigidity theorem of entire spacelike Lagrangian translating solitons in [34] to spacelike translating solitons with general codimensions.…”

mentioning

confidence: 73%

“…From Theorem 2 above, it is easy to see that the corresponding translating vector must be spacelike. In [8], Chen-Qiu proved a nonexistence theorem for complete spacelike translating solitons in R m+n n by establishing a very powerful generalized Omori-Yau maximum principle. They proved that there exists no complete m-dimensional spacelike translating soliton (with a timelike translating vector).…”

Section: Introductionmentioning

confidence: 99%

“…In fact, such nonexistence conclusion still holds for the lightlike translating vector case. Motivating by papers [8,33], we classify m-dimensional complete spacelike translating solitons in R m+n n by affine technique and classical gradient estimates, and obtain the following Bernsterin theorem.…”

Section: Introductionmentioning

confidence: 99%

“…A more precise statement of the assertion in Theorem 3 says that there exists an mdimensional complete spacelike translating soliton in R m+n n only if the translating vector is spacelike. It provides another proof of the nonexistence theorem of translating soliton in [8], and a simple proof of the Bernstein theorem for translating soliton in [33].…”

Section: Introductionmentioning

confidence: 99%

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Rigidity of complete spacelike translating solitons in pseudo-Euclidean space

Liu

2019

Journal of Mathematical Analysis and Applications

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mentioning

confidence: 73%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rigidity of complete spacelike translating solitons in pseudo-Euclidean space

Liu

2019

Journal of Mathematical Analysis and Applications

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“…Using the above Bochner formula and the estimate of V r in [6] (here r denotes the distance function on M), we establish the gradient estimate for V T -harmonic maps.…”

Section: Vt-harmonic Maps From Complete Manifolds Into Geodesic Ballsmentioning

confidence: 99%

On VT-harmonic maps

2019

Self Cite

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V T-harmonic maps generalize the standard harmonic maps, with respect to the structure of both domain and target. These can be manifolds with natural connections other than the Levi-Civita connection of Riemannian geometry, like Hermitian, affine or Weyl manifolds. The standard harmonic map semilinear elliptic system is augmented by a term coming from a vector field V on the domain and another term arising from a 2-tensor T on the target. In fact, this geometric structure then also includes other geometrically defined maps, for instance magnetic harmonic maps. In this paper, we treat V T-harmonic maps and their parabolic analogues with PDE tools. We establish a Jäger-Kaul type maximum principle for these maps. Using this maximum principle, we prove an existence theorem for the Dirichlet problem for V T-harmonic maps. As applications, we obtain results on Weyl/affine/Hermitian harmonic maps between Weyl/affine/Hermitian manifolds, as well as on magnetic harmonic maps from two-dimensional domains. We also derive gradient estimates and obtain existence results for such maps from noncompact complete manifolds.

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Remarks on solitons for inverse mean curvature flow

Kim

Pyo

2020

Mathematische Nachrichten

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Homothetic and translating solitons for the inverse mean curvature flow (IMCF) in ℝ +1 are self-similar solutions deformed by only homothety and translation under IMCF, respectively. In this paper, we address some geometric properties of these solitons. To be specific, the incompleteness for the homothetic soliton of velocity with 0 < < 1 and for any translating soliton are proved. In addition, we show that the quantitative area growth for the homothetic solitons for ≥ 1 and the translating solitons. K E Y W O R D S homothetic soliton, inverse mean curvature flow, translating soliton M S C (2 0 1 0) 53A10, 53C44

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Rigidity of self-shrinkers and translating solitons of mean curvature flows

Cited by 61 publications

References 26 publications

Rigidity of complete spacelike translating solitons in pseudo-Euclidean space

Rigidity of complete spacelike translating solitons in pseudo-Euclidean space

On VT-harmonic maps

Remarks on solitons for inverse mean curvature flow

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