A half-space theorem for graphs of constant mean curvature $0&lt;H&lt;\frac{1}{2}$ in $\mathbb{H}^{2}\times\mathbb{R}$

Mazet, Laurent; Wanderley, Gabriela A.

doi:10.1215/ijm/1455203158

Cited by 5 publications

(1 citation statement)

References 19 publications

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“…However, here the construction of u differs from that carried out in [50], as we use Perron's method instead of an exhaustion argument on compacts Ω j together with the continuity method. This change, inspired by [39,49], allows us to avoid the need of local estimates, necessary to pass to the limit the graphs in Ω j B ε , and directly guarantees the existence of a complete minimal graph to which Theorem 3 will be applied. Define…”

Section: Bernstein and Half-space Properties: Proof Of Theoremmentioning

confidence: 99%

Bernstein and half-space properties for minimal graphs under Ricci lower bounds

Colombo,

Magliaro,

Mari

et al. 2019

Preprint

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In this paper, we prove a new gradient estimate for minimal graphs defined on domains of a complete manifold M with Ricci curvature bounded from below.In particular, we show that positive, entire minimal graphs on manifolds with nonnegative Ricci curvature are constant, and that complete, parabolic manifolds with Ricci curvature bounded from below have the half-space property. We avoid the need of sectional curvature bounds on M by exploiting a form of the Ahlfors-Khas'minskii duality in nonlinear potential theory.

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Section: Bernstein and Half-space Properties: Proof Of Theoremmentioning

confidence: 99%

Bernstein and half-space properties for minimal graphs under Ricci lower bounds

Colombo,

Magliaro,

Mari

et al. 2019

Preprint

View full text Add to dashboard Cite

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Half-Space Theorems for $$\textbf{1}$$-Surfaces of $$\mathbb {H}^3$$

Bessa,

Cruz,

Pessoa

2024

J Geom Anal

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

Kim

Pyo

2021

Proc. Amer. Math. Soc. Ser. B

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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 \} .

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A half-space theorem for graphs of constant mean curvature $0<H<\frac{1}{2}$ in $\mathbb{H}^{2}\times\mathbb{R}$

Abstract: We study a half-space problem related to graphs in H 2 × R, where H 2 is the hyperbolic plane, having constant mean curvature H defined over unbounded domains in H 2 .

Cited by 5 publications

References 19 publications

Bernstein and half-space properties for minimal graphs under Ricci lower bounds

Bernstein and half-space properties for minimal graphs under Ricci lower bounds

Half-Space Theorems for $$\textbf{1}$$-Surfaces of $$\mathbb {H}^3$$

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

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