Rectifiability of sets and measures

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

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, 206 2 / 23

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, 206 3 / 23 i=

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, 206 4 / 23

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, 206 5 / 23

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, 206 6 / 23

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 2 2 + 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, 206 7 / 23

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, 206 8 / 23

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, 206 9 / 23

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, 206 0 / 23

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

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, 206 2 / 23

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, 206 3 / 23

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, 206 4 / 23

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, 206 5 / 23

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, 206 6 / 23

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, 206 7 / 23

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, 206 8 / 23

( ) 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, 206 9 / 23

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, 206 20 / 23

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, 206 2 / 23

( ) 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, 206 22 / 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, 206 23 / 23