A relative isoperimetric inequality for certain warped product spaces

Rafalski, Shawn

doi:10.1515/advgeom-2012-0012

Cited by 2 publications

(2 citation statements)

References 9 publications

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“…7] on R n with density r p , p < 0 , both of which are generalized by this work. We note also that both Montiel [12,13] and Rafalski [21] have obtained related results for graphs over horizontal regions in warped products.…”

Section: Introductionmentioning

confidence: 53%

The Log-Convex Density Conjecture and vertical surface area in warped products

Howe

2015

Advances in Geometry

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Abstract. We examine the vertical component of surface area in the warped product of a Euclidean interval and a fiber manifold with product density. We determine general conditions under which vertical fibers minimize vertical surface area among regions bounding the same volume and use these results to conclude that in many such spaces vertical fibers are isoperimetric. Our main hypothesis is that the surface area of a fiber be a convex function of the volume it bounds. We apply our results in the specific case of R n − {0} realized as the warped product (0, ∞) ×r S n−1 , providing many new examples of densities where spheres about the origin are isoperimetric, including simple densities with finite volume, simple densities that at the origin are neither logconvex nor smooth, and non-simple densities. We also generalize the results of Kolesnikov and Zhdanov on large balls in R n with increasing strictly log-convex simple density. We situate our work in relation to the Log-Convex Density Conjecture of Rosales et al. and the recent work by Morgan, Ritoré, and others on formulating a generalized log-convex density/stable spheres conjecture.

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Section: Introductionmentioning

confidence: 53%

The Log-Convex Density Conjecture and vertical surface area in warped products

Howe

2015

Advances in Geometry

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“…If n = 1, some Bernstein type results for a graph hypersurface in a warped product have been obtained in [1,2,3,4,8,16,18]. In this paper we work in any codimension, and consider the Heinz estimation type problem for the mean curvature.…”

Section: Introductionmentioning

confidence: 99%

Heinz mean curvature estimates in warped product spaces $M\times_{e^ψ}N$

Salavessa

2017

Preprint

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If a graph submanifold (x, f (x)) of a Riemannian warped product space (M m × e ψ N n , g = g + e 2ψ h) is immersed with parallel mean curvature H, then we obtain a Heinz type estimation of the mean curvature. Namely, on each compact domain D) holds, where A ψ (∂D) and V ψ (D) are the ψ-weighted area and volume, respectively. In particular, H = 0 if (M, g) has zero weighted Cheeger constant, a concept recently introduced by D. Impera et al. ([13]). This generalizes the known cases n = 1 or ψ = 0. We also conclude minimality using a closed calibration, assuming (M, g * ) is complete where g * = g + e 2ψ f * h, and for some constants α ≥ δ ≥ 0, C 1 > 0 and β ∈ [0, 1), ∇ * ψ 2 g * ≤ δ, Ricci ψ,g * ≥ α, and det g (g * ) ≤ C 1 r 2β holds when r → +∞, where r(x) is the distance function on (M, g * ) from some fixed point. Both results rely on expressing the squared norm of the mean curvature as a weighted divergence of a suitable vector field.

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A relative isoperimetric inequality for certain warped product spaces

Cited by 2 publications

References 9 publications

The Log-Convex Density Conjecture and vertical surface area in warped products

The Log-Convex Density Conjecture and vertical surface area in warped products

Heinz mean curvature estimates in warped product spaces $M\times_{e^ψ}N$

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