Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow

Proc. Amer. Math. Soc. Ser. B

2021

Self Cite

In this paper, we determine which half-space contains a complete translating soliton of the mean curvature flow and it is related to the well-known half-space theorem for minimal surfaces. We prove that a complete translating soliton does not exist with respect to the velocity v {\mathrm {v}} in a closed half-space H v ~ = { x ∈ R n + 1 ∣ ⟨ x , v ~ ⟩ ≤ 0 } \mathcal {H}_{\widetilde {{\mathrm {v}}}}= \{ x \in \mathbb {R}^{n+1} \mid \langle x, \widetilde {{\mathrm {v}}}\rangle \leq 0 \} for ⟨ v , v ~ ⟩ > 0 \langle {\mathrm {v}}, \widetilde {{\mathrm {v}}} \rangle > 0 , whereas in a half-space H v ~ \mathcal {H}_{\widetilde {{\mathrm {v}}}} , ⟨ v , v ~ ⟩ ≤ 0 \langle {\mathrm {v}}, \widetilde {{\mathrm {v}}} \rangle \leq 0 , a complete translating soliton can be found. In addition, we extend this property to cones: there are no complete translating solitons with respect to v {\mathrm {v}} in a right circular cone C v , a = { x ∈ R n + 1 ∣ ⟨ x ‖ x ‖ , v ⟩ ≤ a > 1 } C_{ {{\mathrm {v}}}, a}=\{ x \in \mathbb {R}^{n+1} \mid \langle \frac {x}{\|x\|} , {{\mathrm {v}}} \rangle \leq a > 1 \} .

show abstract

Section: Example 24 (Translating Bowl and Winglike Translator) Altsmentioning

confidence: 99%

Section: Example 25 (Generalized Winglike Translator)mentioning

confidence: 99%

Half-space type theorem for translating solitons of the mean curvature flow in Euclidean space

Proc. Amer. Math. Soc. Ser. B

2021

Self Cite

show abstract

“…Sato and de Souza Neto [34] classified the zero scalar curvature O( p + 1) × O(q + 1)-invariant hypersurfaces in R p+q+2 for p, q > 1, and analyzed their stability. The authors [27,28] used another different suitable coordinate transformation for the translating soliton to prove the existence of translating solitons for the mean curvature flow and the inverse mean curvature flow and their asymptotic behavior. Then, we completely classify rotationally and birotationally symmetric homothetic solitons for any C > 0, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Then, we completely classify rotationally and birotationally symmetric homothetic solitons for any C > 0, i.e. the self-expander solutions of the IMCF, and verify their asymptotic behavior by following the method used in [1,5,27,28,34] as follows: We first construct the rotationally and birotationally symmetric hypersurfaces and obtain second order nonlinear ordinary differential equations. Secondly, the equations can be changed to systems of first order linear homogeneous ordinary differential equations via the coordinate transformation.…”

Section: Introductionmentioning

confidence: 99%

$\boldsymbol{O(m) \times O(n)}$ -invariant homothetic solitons for inverse mean curvature flow in $\boldsymbol {\mathbb{R}^{m+n}}$

2019

Nonlinearity

Self Cite

The inverse mean curvature flow (IMCF) has been extensively studied not only as a type of geometric flows, but also for its applications to geometric inequalities. The focus is primarily on homothetic solitons for the IMCF in this paper, which are special solutions deformed only homothetically under the flow. We completely classify the profile curves of the higher dimensional rotationally and birotationally symmetric homothetic solitons using the phaseplane analysis. As a characterization of the round cylinder, we obtain that any helicoidal homothetic soliton of IMCF must be round cylinders. In addition, we prove that any surface foliated by circles to be the homothetic soliton of IMCF, which is a cyclic homothetic soliton, is either a surface of revolution or a piece of a round sphere.

show abstract

Translating Solitons for the Inverse Mean Curvature Flow

2019

Results Math