Some calibrated surfaces in manifolds with density

Hieu, Doan The

doi:10.1016/j.geomphys.2011.04.005

Cited by 9 publications

(3 citation statements)

References 10 publications

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“…For more details about Lorentz-Minkowski spaces, maximal surface, calibrations, manifolds with density and Gauss space, we refer the reader to [12], [13], [14], [15], [16], [17], [18] and [19].…”

Section: Introductionmentioning

confidence: 99%

Entire $f$-maximal graphs in the lorentzian product $\Bbb G^n\times\Bbb R_1$

An,

Cuong,

Duyen

et al. 2015

Preprint

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In the Lorentzian product G n × R 1 , we give a comparison theorem between the fvolume of an entire f -maximal graph and the f -volume of the hyperbolic H + r under the condition that the gradient of the function defining the graph is bounded away from 1. This condition comes from an example of non-planar entire f -maximal graph in G n × R 1 and is equivalent to the hyperbolic angle function of the graph being bounded. As a consequence, we obtain a Calabi-Bernstein type theorem for f -maximal graphs in G n × R 1 .

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Section: Introductionmentioning

confidence: 99%

Entire $f$-maximal graphs in the lorentzian product $\Bbb G^n\times\Bbb R_1$

An,

Cuong,

Duyen

et al. 2015

Preprint

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“…For more details about manifolds with density and some relative topics we refer the reader to [2]- [6], [8], [9], [15]- [19], [21].…”

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confidence: 99%

“…The case of −1 < c < 1 (seeFigures 6.3, 6.4, 6.5). If there exists s 0 so that cos[2ξ(s 0 )] = c, then ξ(s) = ξ(s 0 ) is the unique solution of the equation(9) and the corresponding curves are straight lines with the slope −√ 1−c 2 c…”

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confidence: 99%

The classification of constant weighted curvature curves in the plane with a log-linear density

Nam¹

2014

CPAA

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In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in R 2 . The classification gives some interesting phenomena and consequences including: the family of curves converge to a round point when the weighted curvature of curves (or equivalently the forcing term of traveling curved fronts) goes to infinity, a simple proof for a main result in [13] as well as some well-known facts concerning to the isoperimetric problem in the plane with density e y .

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Free boundary stable hypersurfaces in manifolds with density and rigidity results

Castro

Rosales

2014

Journal of Geometry and Physics

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Abstract. Let M be a weighted manifold with boundary ∂M , i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in ∂M . As a consequence, we obtain variational characterizations of critical points and second order minima of the weighted area with or without a volume constraint. Moreover, in the compact case, we obtain topological estimates and rigidity properties for free boundary stable and area-minimizing hypersurfaces under certain curvature and boundary assumptions on M . Our results and proofs extend previous ones for Riemannian manifolds (constant densities) and for hypersurfaces with empty boundary in weighted manifolds.

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Some calibrated surfaces in manifolds with density

Abstract: Abstract. Hyperplanes, hyperspheres and hypercylinders in R n with suitable densities are proved to be weighted area-minimizing by a calibration argument.

Cited by 9 publications

References 10 publications

Entire $f$-maximal graphs in the lorentzian product $\Bbb G^n\times\Bbb R_1$

Entire $f$-maximal graphs in the lorentzian product $\Bbb G^n\times\Bbb R_1$

The classification of constant weighted curvature curves in the plane with a log-linear density

Free boundary stable hypersurfaces in manifolds with density and rigidity results

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