Optimal isoperimetric inequalities

Almgren, Frederick J.

doi:10.1090/s0273-0979-1985-15393-5

Cited by 22 publications

(31 citation statements)

References 5 publications

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“…Let S denote the singular set of B 1 (ξ ∞ ). For any ǫ ∈ (0, 1 2 ], let R ǫ denote the ǫ-regular set of B 1 (ξ ∞ ) defined by…”

Section: Andmentioning

confidence: 99%

“…Clearly, the Sobolev inequality (1.3) is equivalent to the isoperimetric inequality on M . In [1], Almgren proved the sharp isoperimetric inequality for minimizing submanifolds in Euclidean space with arbitrary codimensions. Recently, Brendle [7] proved the sharp isoperimetric inequality for minimal submanifolds in Euclidean space with the codimensions ≤ 2 (see [48] for the relative version).…”

Section: Introductionmentioning

confidence: 99%

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Poincaré inequality on minimal graphs over manifolds and applications

Ding¹

2021

Preprint

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Let B2(p) be an n-dimensional smooth geodesic ball with Ricci curvature ≥ −(n − 1)κ 2 for some κ ≥ 0. We establish the Sobolev inequality and the uniform Neumann-Poincaré inequality on each minimal graph over B1(p) by combining Cheeger-Colding theory and the current theory from geometric measure theory, where the constants in the inequalities only depends on n, κ, the lower bound of the volume of B1(p). As applications, we derive gradient estimates and a Liouville theorem for a minimal graph M over a smooth complete noncompact manifold Σ of nonnegative Ricci curvature and Euclidean volume growth. Furthermore, we can show that any tangent cone of Σ at infinity splits off a line isometrically provided the graphic function of M admits linear growth.

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“…Let S denote the singular set of B 1 (ξ ∞ ). For any ǫ ∈ (0, 1 2 ], let R ǫ denote the ǫ-regular set of B 1 (ξ ∞ ) defined by…”

Section: Andmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Poincaré inequality on minimal graphs over manifolds and applications

Ding¹

2021

Preprint

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“…Inequality (14) is well known. See for example [1], where it was stated for bounded regions in R m . By Faber-Krahn (1) and (14) we obtain the isoperimetric inequality…”

Section: Theorem 2 Suppose T Is a Non-negative Set Function Defined mentioning

confidence: 99%

On the Minimization of Dirichlet Eigenvalues of the Laplace Operator

Berg

Iversen

2011

J Geom Anal

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We study variational problems of the form inf{λ k (Ω) : Ω open in R m , T (Ω) ≤ 1}, where λ k (Ω) is the k'th eigenvalue of the Dirichlet Laplacian acting in L 2 (Ω), and where T is a non-negative set function defined on the open sets in R m , which is invariant under isometries, additive on disjoint families of open sets, and is such that the ball with T (B) = 1 is a minimiser for k = 1. Upper bounds are obtained for the number of components of any bounded minimiser if T satisfies a scaling relation. For example we show that if T is Lebesgue measure and if k ≤ m + 1 then any bounded minimiser has at most 7 components. We also consider variational problems over open sets Ω in R m involving the (m − 1) -dimensional Hausdorff measure of ∂Ω. Mathematics Subject Classification (2000): 49Q10; 49R50; 35P15.

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“…where we used the fact that ∂φ ∂θ = ψ ≥ 1. In view of the similarity of (8) to the formulas for the area and arc length in polar coordinates, one can easily construct a curveγ i from p 1 i to p 2 i parametrized by the angle parameter φ such that l(f −1 (φ)) = dist(p,γ i (φ)) and…”

Section: Stationary Varifoldsmentioning

confidence: 99%

Sharp isoperimetric inequalities for stationary varifolds and area minimizing flat chains mod $k$

Choe¹

1996

Kodai Math. J.

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It is well known [4, 5, 8] that a smooth minimal surface Σ spanning a rectifiable Jordan curve C satisfies the isoperimetric inequality 4πArea(Σ) ≤ Length(C) 2 , (1) where equality holds if and only if C is a circle and Σ is a disk. Some smooth minimal surfaces in R 3 can be physically realized as soap films. However, the soap films formed by dipping some wire frames in soap solution contain interior singular curves. Here arises the main question of this paper: Does (1) still hold for this soap-film-like surface with singularities? In 1986 one of Almgren's results [2] answered this question affirmatively for area minimizing flat chains mod k. In this paper we extend his two-dimensional result and show that (1) holds also for two-dimensional stationary varifolds with connected boundary of multiplicity ≥ 1 (Theorem 2). Moreover, if the bounding curve C consists of k curves having the same end points, we obtain a new type of sharp isoperimetric inequality for area minimizing flat chains mod k spanned by C. Here, unlike (1), equality holds only for the union of k flat half disks with a common diameter (Theorem 3).

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Optimal isoperimetric inequalities

Cited by 22 publications

References 5 publications

Poincaré inequality on minimal graphs over manifolds and applications

Poincaré inequality on minimal graphs over manifolds and applications

On the Minimization of Dirichlet Eigenvalues of the Laplace Operator

Sharp isoperimetric inequalities for stationary varifolds and area minimizing flat chains mod $k$

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