Fine properties of the curvature of arbitrary closed sets

Santilli, Mario

doi:10.1007/s10231-019-00926-w

Cited by 10 publications

(9 citation statements)

References 19 publications

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“…x ∈ S(A, r) ∩ A τ and for every r > 0 by [San19a, 2.13(1)] and [Fed69, 2.10.19(4)], and H n−1 (S(A, r) ∼ R(A)) = 0 for L 1 a.e. r > 0 by [San19a,3.15]. Then Claim 2 follows from 2.9 and Claim 1.…”

Section: Theoremmentioning

confidence: 87%

“…Besides the concept of approximate second fundamental form, in this paper we make use of a more general notion of second fundamental form introduced in [San19a] that can be associated to arbitrary closed sets. The theory of curvature for arbitrary closed sets has been developed in [Sta79], [HLW04], [San19a] and here we summarize those concepts that are relevant for our purpose in the present paper.…”

Section: Curvature For Arbitrary Closed Setsmentioning

confidence: 99%

“…In [San19a] we have systematically investigated a notion of curvature for arbitrary closed sets. The first goal of the present paper is to employ this machinery to analyze the pointwise curvature properties of (m, h) sets.…”

Section: Introductionmentioning

confidence: 99%

“…This condition has deep consequences on the curvature properties of these varieties. For example it implies that the first m principal curvatures of an (m, h) set are finite; this is in sharp contrast with the typical behavior of a convex surface; see [Sch15] and [San19a,6.3].…”

Section: Introductionmentioning

confidence: 99%

“…Remark. It is in general not possible to replace N (A)|R with N (A)|S in the conclusion, even if S is the boundary of a C 1,α convex set A; see the example in[San19a, 6.3].…”

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confidence: 99%

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Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature

Santilli

2020

Bull. Math. Sci.

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In this paper we deal with singular varieties of bounded mean curvature in the viscosity sense. They contain all varifolds of bounded generalized mean curvature. In the first part we investigate the second-order properties of these varieties, obtaining results that are new also in the varifold's setting. In particular we prove that the generalized normal bundle of these varieties satisfies a natural Lusin (N) condition, which allows to extend the classical Coarea formula for the Gauss map of smooth varieties, and to introduce for all integral varifolds of bounded mean curvature a natural definition of second fundamental form, whose trace equals the generalized varifold mean curvature. In the second part, we use this machinery to extend a sharp geometric inequality of Almgren to all compact varieties of bounded mean curvature in the viscosity sense and we characterize the equality case. As a consequence we formulate sufficient conditions to conclude that the area-blow-up set is empty for sequences of varifolds whose first variation is controlled.MSC-classes 2010. 49Q20, 49Q10, 53A07, 53C24, 35D40.Keywords. Bounded mean curvature, varifolds, generalized second fundamental form, generalized Gauss map, Almgren sphere theorem, area blow-up set.1.2 Definition. Suppose A ⊆ R n is a closed set, Ω ⊆ R n is an open set and 1 ≤ m < n is an integer. We say that N (A) satisfies the m dimensional Lusin (N) condition in Ω if and only if the following property holds:is the set of points where A can be touched by a ball from n − m linearly independent directions, (see 2.5).2 In this paper we adopt the terminology in [Alm86, Appendix C] for varifolds; in particular note that the variation function h(V, ·) (i.e. generalized mean curvature of V ) differs from the one adopted in Allard's paper [All72, 4.2] by a sign.

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Section: Theoremmentioning

confidence: 87%

Section: Curvature For Arbitrary Closed Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Remark. It is in general not possible to replace N (A)|R with N (A)|S in the conclusion, even if S is the boundary of a C 1,α convex set A; see the example in[San19a, 6.3].…”

mentioning

confidence: 99%

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Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature

Santilli

2020

Bull. Math. Sci.

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Uniqueness of Critical Points of the Anisotropic Isoperimetric Problem for Finite Perimeter Sets

Rosa

Kolasiński

Santilli

2020

Arch Rational Mech Anal

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Regularity of the distance function from arbitrary closed sets

Kolasiński

Santilli

2022

Math. Ann.

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We investigate the distance function $$\varvec{\delta }_{K}^{\phi }$$ δ K ϕ from an arbitrary closed subset K of a finite-dimensional Banach space $$ (\mathbf {R}^{n}, \phi ) $$ ( R n , ϕ ) , equipped with a uniformly convex $$ \mathscr {C}^{2} $$ C 2 -norm $$ \phi $$ ϕ . These spaces are known as Minkowski spaces and they are one of the fundamental spaces of Finslerian geometry (see Martini et al. in Expo Math 19:97–142, 2001, 10.1016/S0723-0869(01)80025-6). We prove that the gradient of $$\varvec{\delta }_{K}^{\phi }$$ δ K ϕ satisfies a Lipschitz property on the complement of the $$\phi $$ ϕ -cut-locus of K (a.k.a. the medial axis of $$\mathbf {R}^{n} {{\,\mathrm{\sim }\,}}K$$ R n ∼ K ) and we prove a structural result for the set of points outside K where $$\varvec{\delta }_{K}^{\phi }$$ δ K ϕ is pointwise twice differentiable, providing an answer to a question raised by Hiriart-Urruty (Am Math Mon 89:456–458, 1982, 10.2307/2321379). Our results give sharp generalisations of some classical results in the theory of distance functions and they are motivated by critical low-regularity examples for which the available results gives no meaningful or very restricted informations. The results of this paper find natural applications in the theory of partial differential equations and in convex geometry.

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Fine properties of the curvature of arbitrary closed sets

Cited by 10 publications

References 19 publications

Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature

Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature

Uniqueness of Critical Points of the Anisotropic Isoperimetric Problem for Finite Perimeter Sets

Regularity of the distance function from arbitrary closed sets

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