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

Kim, Dae-Hwan; Pyo, Juncheol

doi:10.1088/1361-6544/ab272b

Cited by 7 publications

(5 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Yau [42] and Cheng and Yau [6] established for the Laplacian of a function bounded from above on a complete Riemannian manifold with the Ricci curvature bounded from below. Various versions of the Omori-Yau maximum principle have been used to prove the geometric problems in [5,7,8,25,43]. A detailed introduction to various Omori-Yau maximum principles and their applications can be found in [1].…”

Section: Introductionmentioning

confidence: 99%

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

Kim

Pyo

2021

Proc. Amer. Math. Soc. Ser. B

Self Cite

View full text Add to dashboard 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: Introductionmentioning

confidence: 99%

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

Kim

Pyo

2021

Proc. Amer. Math. Soc. Ser. B

Self Cite

View full text Add to dashboard Cite

show abstract

“…Also, he [13] proved the existence of a unique even solution of the homothetic soliton that is a topological hypercylinder. The present authors [18] completely classified rotationally and birotationally symmetric homothetic solitons for IMCF and their asymptotic behavior from a phase‐plane analysis via an appropriate coordinate transformation. We also proved that any cyclic homothetic soliton in

R^{3}

must be either a surface of revolution or a piece of a round sphere.…”

Section: Introductionmentioning

confidence: 99%

“…We also proved that any cyclic homothetic soliton in

R^{3}

must be either a surface of revolution or a piece of a round sphere. In particular, we [18] obtained that every rotationally and birotationally symmetric homothetic soliton for

C < \frac{1}{n}

R^{n + 1}

is geodesically incomplete.…”

Section: Introductionmentioning

confidence: 99%

Remarks on solitons for inverse mean curvature flow

Kim

Pyo

2020

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

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

show abstract

“…Existence of various solitons are proved recently by G. Drugan, H. Lee and G. Wheeler [DLW], G. Huisken and T. Ilmanen [HuI2], K.M. Hui [H1], [H2], and D. Kim and J. Pyo [KP1], [KP2].…”

Section: Introductionmentioning

confidence: 99%

“…However there are no detailed proofs of these results in [DLW]. On the other hand the existence result Theorem 1.1 is proved by D. Kim and J. Pyo [KP2] using phase plane method. In this paper we will give a different and elementary proof of these results.…”

Section: Introductionmentioning

confidence: 99%

Another proof of the existence of homothetic solitons of the inverse mean curvature flow

Hsu

2020

Preprint

View full text Add to dashboard Cite

We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow (cf. [DLW]) in R n × R, n ≥ 2, of the form (r, y(r)) or (r(y), y) where r = |x|, x ∈ R n , is the radially symmetric coordinate and y ∈ R. More precisely for any 1 n < λ < 1 n−1 and µ < 0, we will give a new proof of the existence of a unique solution r(y) ∈ C 2 (µ, ∞) ∩ C([µ, ∞)) of the equation r yy (y) 1+r y (y) 2 = n−1 r(y) − 1+r y (y) 2 λ(r(y)−yr y (y)) , r(y) > 0, in (µ, ∞) which satisfies r(µ) = 0 and r y (µ) = lim yցµ r y (y) = +∞. We also prove that there exist constants y 2 > y 1 > 0 such that r y (y) > 0 for any µ < y < y 1 , r y (y 1 ) = 0, r y (y) < 0 for any y > y 1 , r yy (y) < 0 for any µ < y < y 2 , r yy (y 2 ) = 0 and r yy (y) > 0 for any y > y 2 . Moreover lim y→+∞ r(y) = 0 and lim y→+∞ yr y (y) = 0.

show abstract

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

Cited by 7 publications

References 30 publications

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

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

Remarks on solitons for inverse mean curvature flow

Another proof of the existence of homothetic solitons of the inverse mean curvature flow

Contact Info

Product

Resources

About