Rectifiability of sets and measures

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1 Rectifiability of sets and measures Tatiana Toro University of Washington IMPA February 7, 206 Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, 206 / 23

2 State of affairs I Let µ be a Radon measure in R m such that 0 < θ n µ(b(a, r)) (µ, a) = lim r 0 r n < for µ a.e a R m Then for µ-a.e. a R m every ν Tan(µ, a) is n-uniform with 0 spt ν. Thus to understand the infinitesimal structure µ we need to understand the structure of n-uniform measures. Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

3 Uniformly distributed measures - Kirchheim-Preiss A measure µ on R m is uniformly distributed if there is a function f µ : (0, ) [0, ] such that µ(b(x, r)) = f µ (r) for all x spt µ and all r > 0. f µ (r) < for some r. For µ uniformly distributed in R m, x R m ( and 0 < s r < µ(b(x, r)) 5 m r ) m fµ (s). ( ) s Cover B(x, r) N i= B(z i, s/2) with z i z j s/2. Since {B(z i, s/4)} N i= are disjoint then N ( s 4) m ( 5 4 r) m. µ(b(x, r)) N µ(b(z i, s/2)). Consider 2 cases: µ(b(z i, s/2)) = 0 or µ(b(z i, s/2)) > 0. If µ(b(z i, s/2)) > 0 there is z spt µ B(z i, s/2), and B(z i, s/2) B(z, s). Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23 i=

4 Uniformly distributed measures - Kirchheim-Preiss A measure µ on R m is uniformly distributed if there is a function f µ : (0, ) [0, ] such that µ(b(x, r)) = f µ (r) for all x spt µ and all r > 0. f µ (r) < for some r. For µ uniformly distributed in R m, x R m and 0 < s r < ( µ(b(x, r)) 5 m r ) m fµ (s). s For µ uniformly distributed in R m ( ) f µ (r) < for all r > 0 e s x z 2 dµ(z) = µ(b(x, ln r )) dr converges for s > 0, 0 s x R m. e s x z 2 dµ(z) = e s y z 2 dµ(z) for all x, y spt µ and s > 0. Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

5 Uniformly distributed measures - Kirchheim-Preiss Let µ be uniformly distributed in R m, x 0 spt µ, for s > 0 and x R m ( F (x, s) = e s x z 2 e s x 0 z 2) dµ(z) is well defined and independent of x 0. x spt µ iff F (x, s) = 0 for all s > 0. If x / spt µ there is s x > 0 so that s > s x, F (x, s) < 0 ( ). spt µ is a real analytic variety. There are n {0,,, m}, c > 0 and G R m an open set such that G spt µ is an n-dimensional analytic submanifold of R m. µ(r m \G) = H n (R m \G) = 0. R m \G countable union of analytic submanifolds of R m of dimension less than n. µ(a) = ch n (A G spt µ) = ch n (A spt µ) for A R m Borel. Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

6 State of affairs II Let µ be a Radon measure in R m such that Then: 0 < θ n µ(b(a, r)) (µ, a) = lim r 0 r n < for µ a.e a R m For µ-a.e. a R m every ν Tan(µ, a) is n-uniform thus uniformly distributed. For x G spt ν (n-analytic submanifold) if λ Tan(ν, x) then there exists c > 0 such that λ = ch n V x where V x = T x spt ν x. Since tangents to tangents are tangents λ Tan(µ, a). λ F n,m = F = {ch n V : c > 0, V G(m, n)}, i.e. λ is flat. For µ-a.e. a R m, Tan(µ, a) F. Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

7 Are all n-uniform measures flat? How large can the singular set be? Theorem If ν is n-uniform in R n+, Σ = spt ν then ν = ch n modulo translation and rotation Preiss For n =, 2 Σ = R n {0} Kowalski-Preiss For n 3, Σ and Σ = R n {0}, or Σ = {(x, x 2, x 3, x 4,, x n+ ) R n+ : x4 2 = x 2 + x x 3 2}. Recent work of D. Nimer addresses the question in higher codimensions. Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

8 How do we prove rectifiability of µ? Theorem (Preiss) Let µ be a Radon measure in R m with 0 < θ (µ, n a) θ n, (µ, a) < µ a.e. a R m. Then the following are equivalent: µ is n-rectifiable µ a.e a R m there is V a an n-dimensional space in R m so that Tan(µ, a) = {ch n V a : 0 < c < } µ a.e. a R m, Tan(µ, a) F. Does Tan(µ, a) F = Tan(µ, a) F for µ-a.e. a R m? Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

9 Metric on the space of Radon measures in R m For r (0, ), let L(r) = {f : R m [0, ), spt f B r and Lip f }. For Radon measures Φ and Ψ on R m set { } F r (Φ, Ψ) = sup f dφ f dψ ; f L(r). F r (Φ) = F r (Φ, 0) = (r z ) + dφ(z). ( ) If µ, µ i (i N) Radon measures in R m, then µ i µ iff lim i F r (µ i, µ) = 0 r > 0. Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

10 Cones of measures Let R be the space of Radon measures in R m. For Φ, Ψ R define d(φ, Ψ) = 2 i min{, F i (Φ, Ψ)}. i= (R, d) is a complete separable metric space. ( ) If µ, µ i (i N) Radon measures in R m, then µ i µ iff lim i d(µ i, µ) = 0. M R, 0 M is a cone if cψ M for all Ψ M and c > 0. A cone M is a d-cone if T 0,r [Ψ] M for all Ψ M. For s > 0 the s-distance between a d-cone M and Φ R is { } Φ d s (Φ, M) = inf F s ( F s (Φ), Ψ) : Ψ M and F s(ψ) = if F s (Φ) 0. If F s (Φ) = 0 set d s (Φ, M) =. Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

11 A picture to illustrate d s (Φ, M) M Φ Φ F s(φ) {Ψ R : F s (Ψ) = } { } Φ d s (Φ, M) = inf F s ( F, Ψ) : Ψ M and F s(φ) s(ψ) = Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, 206 / 23

12 Additional properties If M d-cone and Φ R d s (Φ, M) ( ) ds (Φ, M) = d (T 0,s [Φ], M) If µ i µ and F s (µ) > 0 then d s (µ, M) = lim i d s (µ i, M). If µ is a non-zero Radon measure Tan(µ, a) is a d-cone. If ν Tan(µ, a), T 0,r [ν] Tan(µ, a). If µ is a non-zero Radon measure {ν Tan(µ, a) : F (ν) = } is closed under weak convergence (i.e. in the topology induced by d). The basis of a d-cone M is the set {Ψ M : F (Ψ) = }. Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

13 Compact basis Let M be a d-cone in R with a closed basis then M has a compact basis iff there exists κ > such that Ψ(B(0, 2r)) κψ(b(0, r)) for all Ψ M and r > 0,i.e. the doubling constant is uniform on M. Let µ R, a spt µ if c a = lim sup r 0 then Tan(µ, a) has a compact basis. ( ) µ(b(a, 2r)) µ(b(a, r)) <, Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

14 Connectivity properties of Tan(µ, a) - Preiss Let M and F be d-cones in R. Assume that F M, F relatively closed in M and M has a compact basis. Suppose that there exists ɛ 0 > 0 such that if ( ) d r (Φ, F) < ɛ 0 r r 0 > 0 then Φ F. Then for a Radon measure µ and a spt µ if Tan(µ, a) M and Tan(µ, a) F then Tan(µ, a) F Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

15 State of affairs III Let µ be a Radon measure in R m such that 0 < θ n µ(b(a, r)) (µ, a) = lim r 0 r n < for µ a.e a R m Then for µ-a.e. a R m : F M, where F is the set of n-flat measures in R m and M is the set of n-uniform measures in R m with 0 in their support. F is closed in M, which is a d-cone with compact basis. Tan(µ, a) M. Tan(µ, a) F. If ( ) holds then Tan(µ, a) F = µ n-rectifiable Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

16 What does ( ) really mean? Let Φ be n-uniform. Recall { } Φ d r (Φ, F = inf F r ( F r (Φ), Ψ) : Ψ F and F r (Ψ) =. F r (Φ) = (r y ) + dφ = Φ({y : (r y ) + > t}) dt = = r 0 r 0 Φ({y : y < r t})) dt = c(r t) n dt = c r n+ n + r 0 Φ(B r t ) dt d r (Φ, F) = d r (cφ, F) for c = ω n then F r (Φ) = If Ψ = ch n ωnr n+ V, F r (Ψ) = c n+ = implies c = ωnr n+ n+. ( ωnr n+ n+ ). Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

17 d r (Φ, F) = Note that for n + ω n r n+ inf V G(m,n) f dφ n + ω n r n+ sup f L(r) f (x) = dist 2 (r x )+ (x, V ) f dh n r 2 f dφ L(r). f dh n V V n + ω n r n+ f dφ n + ω n r n+ dist 2 (x, V ) r B r/2 2r 2 dφ n + 2ω n r n+2 dist 2 (x, V ) dφ B r/2 If ( ) d r (Φ, F) < ɛ 0 for r r 0 > 0, then n + inf V G(m,n) ω n r n+2 dist 2 (x, V ) dφ < 2ɛ 0. B r/2 Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

18 The condition d r (Φ, F) < ɛ 0 r r 0 implies that there exists ɛ > 0 such that inf V G(m,n) ω n r n+2 dist 2 (x, V ) dφ < ɛ for r r 0. B r Thus for r r 0 there exists V r G(m, n) such that ( ) ω n r n+2 dist 2 (x, V r ) dφ ɛ B r To prove ( ) we need to show that if Φ n-uniform is close to flat at infinity in L 2 as in ( ) then Φ is flat. Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

19 ( ) yields geometric information There is a large good subset G r of spt Φ in B r which is close tov r By Chebyshev s inequality if ( ) holds then satisfies G r = {x spt Φ B r : dist (x, V r ) ɛ /4 r} Φ(B r \G r ) ɛ /2 Φ(B r ). r V r G r Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

20 The small subset of spt Φ inb r \G r is also close tov r r V r y spt Φ Let r = r( 2ɛ /2n ). If y (B r \G r ) spt Φ, ρ = min{dist (y, G r ), dist (y, B r )} > 0 then B ρ (y) B r \G r ω n ρ n = Φ(B ρ (y)) Φ(B r \G r ) ɛ /2 Φ(B r ) = ω n ɛ /2 r n. Thus ρ = dist (y, G r ) ɛ /2n r. For y (B r( 2ɛ /2n ) \G r ) spt Φ, dist (y, V r ) dist (y, G r ) + dist (G r, V r ) ɛ /2n r. Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

21 Summary If d r (Φ, F) < ɛ 0 for r r 0 > 0 there is V r such that ( ) ω n r n+2 dist 2 (x, V r ) dφ ɛ B r with ɛ = C(n)ɛ 0. In this case for y B r( 2ɛ /2n ) spt Φ, dist (y, V r ) ɛ /2n r. spt Φ close tov r in L 2 sense in B r then spt Φ close V r in L sense in B r( 2ɛ /2n ) Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

22 ( ) yields additional geometric information for Φ n-uniform Not only is spt Φ close to V r but V r is also close to spt Φ. V τ spt Φ Can there be holes of size τ? NO if ɛ is small enough. Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

23 ( ) yields additional geometric information for Φ n-uniform Lemma: Given τ > 0 there exists ɛ = ɛ (τ, n, m) > 0 such that if Φ is n-uniform in R m with 0 spt Φ, Φ(B ) = and for some V G(m, n) B dist 2 (x, V ) dφ < ɛ, then for all z V B B(z, τ) spt Φ. Tatiana Toro (University of Washington) Structure of n-uniform measure in R m February 7, / 23

Cones of measures. Tatiana Toro. University of Washington. Quantitative and Computational Aspects of Metric Geometry

Cones of measures. Tatiana Toro. University of Washington. Quantitative and Computational Aspects of Metric Geometry Cones of measures Tatiana Toro University of Washington Quantitative and Computational Aspects of Metric Geometry Based on joint work with C. Kenig and D. Preiss Tatiana Toro (University of Washington)