On minimal surfaces in a class of Finsler 3-spheres

Cui, Ningwei

doi:10.1007/s10711-012-9819-9

Cited by 13 publications

(14 citation statements)

References 11 publications

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“…Remark If

\overset{W}{true}

is a Killing vector field of constant length, then χ is constant both in the BH‐case and the HT‐case, and the formula in Theorem reduces to the same one as in Corollary , which is Theorem 3.3 in . This enables us to get both the nontrivial BH‐minimal surfaces and the HT‐minimal surfaces in the Bao–Shen sphere in .…”

Section: Mean Curvature Of Submanifolds By the Methods Of Navigationmentioning

confidence: 91%

“…whereỸ α =h αβỸ β . (See the proof in Page 93 in [4]). For the index 1 ≤ α ≤ n + 1, recalling that W = W j ∂ ∂x j and W j = h ijW β z β i , and noticing the notation (7), we compute…”

Section: Mean Curvature Of Submanifolds By the Methods Of Navigationmentioning

confidence: 98%

“…For the induced Randers metric F , the function

scriptF

can be simultaneously given by

F = \frac{ϱ}{χ} \sqrt{d e t (h_{i j})},

where

ϱ = ϱ (s)

and χ are given by

\begin{matrix} true \\ right (ϱ, χ) = ({, \begin{matrix} ({frakturs}^{- \frac{n}{2}}, 1), for the BH case, \\ ({frakturs}^{\frac{1}{2}}, {(1 - false∥ \overset{W}{true} ∥_{\overset{h}{true}}^{2})}^{\frac{n + 1}{2}}), for the HT case . \end{matrix}) \end{matrix}

See Section in for the details. The volume ratio function for the Randers metric introduced in is given by

\begin{matrix} true \\ right normalΦ false(frakturs false) & : = & left 2 ϱ^{'} (s) (1 - s) + ϱ (s) \\ = & left \{\begin{matrix} right {frakturs}^{- \frac{n}{2}} (() - n s^{- 1} + n + 1), for ... \end{matrix} \end{matrix}

…”

Section: Mean Curvature Of Submanifolds By the Methods Of Navigationmentioning

confidence: 99%

“…Proof First, for any vector field

\tilde{Y} = {\overset{Y}{true}}^{α} \partial_{α} \in T \tilde{M}

along f , we have

〈\overset{Y}{true}, N〉 N = {\overset{Y}{true}}_{α} ({\tilde{h}}^{α γ} - B^{α γ}) \partial_{γ},

where

{\overset{Y}{true}}_{α} = {\overset{h}{true}}_{α β} {\overset{Y}{true}}^{β}

. (See the proof in Page 93 in .) For the index

1 \leq α \leq n + 1

, recalling that

W = W^{j} \frac{\partial}{\partial x^{j}}

and

W^{j} = h^{i j} {\overset{W}{true}}_{β} z_{i}^{β}

, and noticing the notation , we compute

{([], \nabla_{d f (W)}^{\overset{M}{true}}, \tilde{W})}_{α} = {([], W^{j}, z_{j}^{η}, \nabla_{\partial_{η}}^{\overset{M}{true}}, \tilde{W})}_{α} = W^{j} z_{j}^{η} {\overset{W}{true}}_{α | η} = B^{η β} {\overset{W}{true}}_{β} {\overset{W}{true}}_{α | η},

which implies the first equation of immediately.…”

Section: Mean Curvature Of Submanifolds By the Methods Of Navigationmentioning

confidence: 99%

“…The author in gave a formula of the mean curvature form in the Randers space by using the navigation data

(() \tilde{h}, \tilde{W})

in the case that

\overset{W}{true}

is a Killing vector field of constant length. However, according to the classification of Randers space forms in , the hyperbolic Randers space forms are modeled on the nonpositvely curved Riemannian space forms with conformal (Killing) vector fields.…”

Section: Introductionmentioning

confidence: 99%

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Nontrivial minimal surfaces in a hyperbolic Randers space

Cui

Shen

2016

Mathematische Nachrichten

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The contribution of this paper is two‐fold. The first one is to derive a simple formula of the mean curvature form for a hypersurface in the Randers space with a Killing vector field, by considering the Busemann–Hausdorff measure and Holmes–Thompson measure simultaneously. The second one is to obtain the explicit local expressions of two types of nontrivial rotational BH‐minimal surfaces in a Randers domain of constant flag curvature K=−1, which are the first examples of BH‐minimal surfaces in the hyperbolic Randers space.

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“…Remark If

\overset{W}{true}

Section: Mean Curvature Of Submanifolds By the Methods Of Navigationmentioning

confidence: 91%

Section: Mean Curvature Of Submanifolds By the Methods Of Navigationmentioning

confidence: 98%

“…For the induced Randers metric F , the function

scriptF

can be simultaneously given by

F = \frac{ϱ}{χ} \sqrt{d e t (h_{i j})},

where

ϱ = ϱ (s)

and χ are given by

\begin{matrix} true \\ right (ϱ, χ) = ({, \begin{matrix} ({frakturs}^{- \frac{n}{2}}, 1), for the BH case, \\ ({frakturs}^{\frac{1}{2}}, {(1 - false∥ \overset{W}{true} ∥_{\overset{h}{true}}^{2})}^{\frac{n + 1}{2}}), for the HT case . \end{matrix}) \end{matrix}

See Section in for the details. The volume ratio function for the Randers metric introduced in is given by

\begin{matrix} true \\ right normalΦ false(frakturs false) & : = & left 2 ϱ^{'} (s) (1 - s) + ϱ (s) \\ = & left \{\begin{matrix} right {frakturs}^{- \frac{n}{2}} (() - n s^{- 1} + n + 1), for ... \end{matrix} \end{matrix}

…”

Section: Mean Curvature Of Submanifolds By the Methods Of Navigationmentioning

confidence: 99%

“…Proof First, for any vector field

\tilde{Y} = {\overset{Y}{true}}^{α} \partial_{α} \in T \tilde{M}

along f , we have

〈\overset{Y}{true}, N〉 N = {\overset{Y}{true}}_{α} ({\tilde{h}}^{α γ} - B^{α γ}) \partial_{γ},

where

{\overset{Y}{true}}_{α} = {\overset{h}{true}}_{α β} {\overset{Y}{true}}^{β}

. (See the proof in Page 93 in .) For the index

1 \leq α \leq n + 1

, recalling that

W = W^{j} \frac{\partial}{\partial x^{j}}

and

W^{j} = h^{i j} {\overset{W}{true}}_{β} z_{i}^{β}

, and noticing the notation , we compute

{([], \nabla_{d f (W)}^{\overset{M}{true}}, \tilde{W})}_{α} = {([], W^{j}, z_{j}^{η}, \nabla_{\partial_{η}}^{\overset{M}{true}}, \tilde{W})}_{α} = W^{j} z_{j}^{η} {\overset{W}{true}}_{α | η} = B^{η β} {\overset{W}{true}}_{β} {\overset{W}{true}}_{α | η},

which implies the first equation of immediately.…”

Section: Mean Curvature Of Submanifolds By the Methods Of Navigationmentioning

confidence: 99%

“…The author in gave a formula of the mean curvature form in the Randers space by using the navigation data

(() \tilde{h}, \tilde{W})

in the case that

\overset{W}{true}

Section: Introductionmentioning

confidence: 99%

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Nontrivial minimal surfaces in a hyperbolic Randers space

Cui

Shen

2016

Mathematische Nachrichten

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On Minimal Surfaces Immersed in Three Dimensional Kropina Minkowski Space

2021

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In this paper we consider a three dimensional Kropina space and obtain a partial differential equation that characterizes minimal surfaces with the induced metric. Using this characterization equation we study various immersions of minimal surfaces. In particular, we obtain the partial differential equation that characterizes the minimal translation surfaces and show that the plane is the only such surface.

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