Geometric Sobolev-like embedding using high-dimensional Menger-like curvature

Abstract

Abstract. We study a modified version of Lerman-Whitehouse Menger-like curvature defined for (m + 2) points in an n-dimensional Euclidean space. For 1 ≤ l ≤ m + 2 and an m-dimensional set Σ ⊂ R n we also introduce global versions of this discrete curvature, by taking supremum with respect to (m+2−l) points on Σ. We then define geometric curvature energies by integrating one of the global Menger-like curvatures, raised to a certain power p, over all l-tuples of points on Σ. Next, we prove that if Σ is compact and m-Ahlfors regular and if p is greater than the dimension of the set of all l-tuples of points on Σ (i.e. p > ml), then the P. Jones' β-numbers of Σ must decay as r τ with r → 0 for some τ ∈ (0, 1). If Σ is an immersed C 1 manifold or a bilipschitz image of such set then it follows that it is Reifenberg flat with vanishing constant, hence (by a theorem of David, Kenig and Toro) an embedded C 1,τ manifold. We also define a wide class of other sets for which this assertion is true. After that, we bootstrap the exponent τ to the optimal one α = 1 − ml/p showing an analogue of the Morrey-Sobolev embedding theorem W 2,p ⊆ C 1,α . Moreover, we obtain a qualitative control over the local graph representations of Σ only in terms of the energy.

Sufficient conditions for C^1,α parametrization and rectifiability

“…Several attempts have been made to prove similar analogues for sets (or measures) of dimension more than 1, see for instance [Paj96b,Paj96a]. Menger curvature was also introduced to attempt to characterize rectifiability (see, among others, [Lég99,LW11,LW09,KS13,BK12,Kol15,Meu18,Goe18,GG19]). Other approaches can be found in [Mer16,Del08,San19]).…”

“…In [Kol15], a bound on Menger curvature together with other regularity assumptions leads to a pointwise bound on β-numbers: this is the same bound which appears in [DKT01]. If in addition the set is fine, which among other things implies Reifenberg flatness allowing for small holes (that is, at scale r holes are of the size of β E ∞ (x, r)), then the same conclusion as in [DKT01] holds, that is, the set can be parametrized by a C 1,α map.…”

“…It is interesting to note that in [DKT01] Reifenberg flatness, which does not allow for any holes, is used. On the other hand, in [Kol15] they allow small holes, that is, of size bounded by β. In contrast, we only require the set to be one-sided Reifenberg flat, which does not impose any restrictions on the size of the holes.…”

How averaged Menger curvatures control regularity and topology of curves and surfaces

“…with straight nodal lines such as the graph of the smooth function f (x, y) := xy, where the inverse of the circumsphere radius as a possible integrand is not bounded. This led us to the idea to use a very similar but less singular integrand defined on tetrahedra T = (ξ, x, y, z) of surface points, namely Volume(T )/(Area(T ) · (diam T ) 2 ) (Strzelecki & von der Mosel (2011)), where Area(T ) is the sum of all facet areas of T , or even simpler (Kolasiński (2011(Kolasiński ( , 2012), and for all dimensions k < n and (k + 1)-dimensional simplices T = (x 1 , . .…”

“…Again, for exponents above the scale invariant case we obtain a geometric Morrey-Sobolev embedding theorem (Strzelecki & von der Mosel (2011);Kolasiński (2011Kolasiński ( , 2012):…”