Normal and Integral Currents

Federer, Herbert; Fleming, Wendell H.

doi:10.2307/1970227

Cited by 802 publications

(668 citation statements)

References 13 publications

Supporting

Mentioning

654

Contrasting

Unclassified

Order By: Relevance

“…for f ∈ W 1,p (R n ), 1 ≤ p < n, see Aubin [4] and Talenti [42] for 1 < p < n, Federer and Fleming [17] and Maz'ya [37] for p = 1. This can be seen from…”

Section: Introductionmentioning

confidence: 99%

Note on Affine Gagliardo–Nirenberg Inequalities

Zhai

2010

Potential Anal

View full text Add to dashboard Cite

Abstract. This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine L p −Sobolev inequalities. The logarithmic version of affine L p −Sobolev inequalities is verified. Moreover, An alternative proof of the affine Moser-Trudinger and Morrey-Sobolev inequalities is given. The main tools are the equimeasurability of rearrangements and the strengthened version of the classical Pólys-Szegö principle.

show abstract

“…for f ∈ W 1,p (R n ), 1 ≤ p < n, see Aubin [4] and Talenti [42] for 1 < p < n, Federer and Fleming [17] and Maz'ya [37] for p = 1. This can be seen from…”

Section: Introductionmentioning

confidence: 99%

Note on Affine Gagliardo–Nirenberg Inequalities

Zhai

2010

Potential Anal

View full text Add to dashboard Cite

show abstract

“…In this direction we note that Federer and Fleming [9] showed that any non-trivial integral homology class in H p (M, Z)) corresponds to a stable current. Later Lawson and Simons [15] showed that there are no stable integral currents in the sphere S n , and there is no integral current in a submanifold M m of S n when the second fundamental form of M m satisfies a pinching condition.…”

Section: Introductionmentioning

confidence: 85%

“…Lemma 3.1 ( [9], [16], [19]). Let M n be a compact, n-dimensional submanifold of the space formM (c) of curvature c ≥ 0 with second fundamental form h, and let p, q be positive integers such that p + q = n. If for any x ∈ M n and any orthonormal basis {e 1 , .…”

Section: Homology Of Contact Cr-warped Submanifoldsmentioning

confidence: 99%

Homology of Contact Cr-Warped Product Submanifolds of an Odd-Dimensional Unit Sphere

Şahin¹,

Şahin²

2015

Bulletin of the Korean Mathematical Society

View full text Add to dashboard Cite

show abstract

“…The mathematical study of various forms of isoperimetric inequalities has a substantial history; see, for example, the survey article [OR]. The first leastvolume isoperimetric inequalities in general dimensions and codimensions of which I know appeared in [FF,Corollary 6.3] (with nonoptimal constants depending on dimension and codimension). The size function (necessary for real multiplicity currents) is first used in [A2].…”

Section: M(t) • S(t) 1^m /M(q) Is Minimizedmentioning

confidence: 99%

Optimal isoperimetric inequalities

Almgren¹

1985

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

It is well known (and true) that among all n-dimensional closed surfaces (boundaries of bounded regions) in i2 n+1 having n-area equal to that of the standard n-sphere dB n+1 (0,1), the n-sphere itself uniquely (up to translations and sets of measure 0) encloses the largest volume. An equivalent formulation of this statement is the optimal isoperimetric inequality which asserts thatfor any nonempty bounded region Q in i2 n+1 with equality if and only if (up to sets of measure 0) Q = int B n+1 (p, r) for some p € # n+1 and 0 < r < oo; here £ n+1 denotes (n + l)-dimensional Lebesgue measure, M n denotes ndimensional Hausdorff measure, B n+1 (p,r) = R n+1 fl {x: \x -p\ < r}, and 7(n + 1) = £ n+1 (B n+1 (0, l))/[>/ n (a5 n+1 (0, l))]^1)/* is the optimal isoperimetric constant. Q need not be the only bounded region having dQ as boundary.We announce several new optimal isoperimetric inequalities of geometric measure theory, proved in [Al], which are valid in general dimensions and codimensions.

show abstract

Normal and Integral Currents

Cited by 802 publications

References 13 publications

Note on Affine Gagliardo–Nirenberg Inequalities

Note on Affine Gagliardo–Nirenberg Inequalities

Homology of Contact Cr-Warped Product Submanifolds of an Odd-Dimensional Unit Sphere

Optimal isoperimetric inequalities

Contact Info

Product

Resources

About