Geometry of uniform measures A. Dali Nimer (University of - - PowerPoint PPT Presentation

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Geometry of uniform measures

• A. Dali Nimer

(University of Washington) GMT workshop at Warwick July 12, 2017

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Preiss’s Theorem

Let 0 < n ≤ d < ∞ be integers, and Φ be a Radon measure in Rd. If Φ is such that 0 < Θn(Φ, x) := lim

r→0

Φ(B(x, r)) ωnr n < ∞ for Φ-almost every x ∈ Rd. Then Φ ≪ Hn and Φ-almost all of Rd can be covered by a countable union of continuously differentiable n-submanifolds, i.e. Φ is n-rectifiable.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Preiss’s Theorem

Let 0 < n ≤ d < ∞ be integers, and Φ be a Radon measure in Rd. If Φ is such that 0 < Θn(Φ, x) := lim

r→0

Φ(B(x, r)) ωnr n < ∞ for Φ-almost every x ∈ Rd. Then Φ ≪ Hn and Φ-almost all of Rd can be covered by a countable union of continuously differentiable n-submanifolds, i.e. Φ is n-rectifiable.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Uniformly distributed and n-uniform measures

Let µ be a Radon measure in Rd. We say µ is uniformly distributed if there exists a function h : R+ → R+ such that for all x ∈ spt(µ), r > 0: µ(B(x, r)) = h(r). If there exists c > 0 such that h(r) = cr n, i.e. for all x ∈ spt(µ), r > 0: µ(B(x, r)) = cr n, we say µ is n-uniform.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Uniformly distributed and n-uniform measures

Let µ be a Radon measure in Rd. We say µ is uniformly distributed if there exists a function h : R+ → R+ such that for all x ∈ spt(µ), r > 0: µ(B(x, r)) = h(r). If there exists c > 0 such that h(r) = cr n, i.e. for all x ∈ spt(µ), r > 0: µ(B(x, r)) = cr n, we say µ is n-uniform.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Facts about n-uniform measures I

The only d-uniform measures in Rd are multiples of Lebesgue measure. The 1-uniform measures in Rd, d ≥ 1, are multiples of H1L, for some line L. The 2-uniform measures in Rd, d ≥ 2, are multiples of H2P, for some 2-plane P. [Preiss]

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Facts about n-uniform measures I

The only d-uniform measures in Rd are multiples of Lebesgue measure. The 1-uniform measures in Rd, d ≥ 1, are multiples of H1L, for some line L. The 2-uniform measures in Rd, d ≥ 2, are multiples of H2P, for some 2-plane P. [Preiss]

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Facts about n-uniform measures I

The only d-uniform measures in Rd are multiples of Lebesgue measure. The 1-uniform measures in Rd, d ≥ 1, are multiples of H1L, for some line L. The 2-uniform measures in Rd, d ≥ 2, are multiples of H2P, for some 2-plane P. [Preiss]

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Example of a non flat 3-uniform measure

Let C ⊂ R4 be the following set: C = {(x1, x2, x3, x4) ∈ R4; x2

4 = x2 1 + x2 2 + x2 3}.

The measure H3C is 3-uniform. [Preiss]

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Example of a non flat 3-uniform measure

Let C ⊂ R4 be the following set: C = {(x1, x2, x3, x4) ∈ R4; x2

4 = x2 1 + x2 2 + x2 3}.

The measure H3C is 3-uniform. [Preiss]

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Facts about uniform measures II

The support of an n-uniform measure in Rn+1 can only be an n-plane or (up to rotation) Rn−3 × C. [Kowalski-Preiss] An n-uniform measure is either flat or “far from flat”.[Preiss] The support of a uniformly distributed measure is an analytic variety [Kirchheim-Preiss]

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Facts about uniform measures II

The support of an n-uniform measure in Rn+1 can only be an n-plane or (up to rotation) Rn−3 × C. [Kowalski-Preiss] An n-uniform measure is either flat or “far from flat”.[Preiss] The support of a uniformly distributed measure is an analytic variety [Kirchheim-Preiss]

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Facts about uniform measures II

The support of an n-uniform measure in Rn+1 can only be an n-plane or (up to rotation) Rn−3 × C. [Kowalski-Preiss] An n-uniform measure is either flat or “far from flat”.[Preiss] The support of a uniformly distributed measure is an analytic variety [Kirchheim-Preiss]

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Conical n-uniform measures

Approach to the problem: consider conical n-uniform measures. An n-uniform measure ν is said to be conical if for all Borel A ⊂ Rd and r > 0 ν(rA) = r nν(A). Reduce the study of such measures to the study of their spherical component Ω = spt(ν) ∩ Sd−1.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Conical n-uniform measures

Approach to the problem: consider conical n-uniform measures. An n-uniform measure ν is said to be conical if for all Borel A ⊂ Rd and r > 0 ν(rA) = r nν(A). Reduce the study of such measures to the study of their spherical component Ω = spt(ν) ∩ Sd−1.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Conical n-uniform measures

Approach to the problem: consider conical n-uniform measures. An n-uniform measure ν is said to be conical if for all Borel A ⊂ Rd and r > 0 ν(rA) = r nν(A). Reduce the study of such measures to the study of their spherical component Ω = spt(ν) ∩ Sd−1.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Theorem: spherical component as a uniform measure

Suppose ν an n-uniform conical measure in Rd. Let σ be its spherical component i.e. σ = Hn−1Ω where Ω = spt(ν) ∩ Sd−1. Then σ is uniformly distributed: for all x ∈ Ω, for all 0 ≤ r ≤ 2 σ(B(x, r)) = Hn−1 B(e, r) ∩ Sn−1 , where e ∈ Sn−1 is arbitrarily chosen.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Archimedes’s Theorem

For every e ∈ S2, and 0 < r ≤ 2, we have: H2(B(e, r) ∩ S2) = πr 2.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Consequence for conical 3-uniform measures

ν conical 3-uniform measure in Rd, σ the spherical component of ν. Then σ is 2-uniform i.e for x ∈ spt(σ), for 0 < r ≤ 2 σ(B(x, r)) = πr 2.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Consequence for conical 3-uniform measures

ν conical 3-uniform measure in Rd, σ the spherical component of ν. Then σ is 2-uniform i.e for x ∈ spt(σ), for 0 < r ≤ 2 σ(B(x, r)) = πr 2.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Characterization of conical 3-uniform measures

Ω ⊂ Sd−1 and σ = H2Ω is 2-uniform (up to r = 2). Then Ω =

• ∪2M

i=1Si

• ,

(1) where the Si’s are mutually disjoint 2-spheres. The Si’s must also satisfy certain “rigidity” conditions.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Characterization of conical 3-uniform measures

Ω ⊂ Sd−1 and σ = H2Ω is 2-uniform (up to r = 2). Then Ω =

• ∪2M

i=1Si

• ,

(1) where the Si’s are mutually disjoint 2-spheres. The Si’s must also satisfy certain “rigidity” conditions.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

A family of 3-uniform measures I

Assume Ω is a union of spheres in R2(m+1) such that Ω =

2m

i=1 Si

• where:

1 {αi}2m

i=1 is a distance-symmetric set of points (up to

normalization of distances) in a linear 2m − 1-plane W .

2 For all i, Si ⊂ V + αi where V = W ⊥ is a linear 3-plane and

Si is the 2- sphere of radius

1 √ 2m and center αi.

Then σ(B(x, r)) = πr 2, for x ∈ Ω , 0 ≤ r ≤ 2. (2)

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

A family of 3-uniform measures I

Assume Ω is a union of spheres in R2(m+1) such that Ω =

2m

i=1 Si

• where:

1 {αi}2m

i=1 is a distance-symmetric set of points (up to

normalization of distances) in a linear 2m − 1-plane W .

2 For all i, Si ⊂ V + αi where V = W ⊥ is a linear 3-plane and

Si is the 2- sphere of radius

1 √ 2m and center αi.

Then σ(B(x, r)) = πr 2, for x ∈ Ω , 0 ≤ r ≤ 2. (2)

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

A family of 3-uniform measures I

Assume Ω is a union of spheres in R2(m+1) such that Ω =

2m

i=1 Si

• where:

1 {αi}2m

i=1 is a distance-symmetric set of points (up to

normalization of distances) in a linear 2m − 1-plane W .

2 For all i, Si ⊂ V + αi where V = W ⊥ is a linear 3-plane and

Si is the 2- sphere of radius

1 √ 2m and center αi.

Then σ(B(x, r)) = πr 2, for x ∈ Ω , 0 ≤ r ≤ 2. (2)

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

A family of 3-uniform measures I

Assume Ω is a union of spheres in R2(m+1) such that Ω =

2m

i=1 Si

• where:

1 {αi}2m

i=1 is a distance-symmetric set of points (up to

normalization of distances) in a linear 2m − 1-plane W .

2 For all i, Si ⊂ V + αi where V = W ⊥ is a linear 3-plane and

Si is the 2- sphere of radius

1 √ 2m and center αi.

Then σ(B(x, r)) = πr 2, for x ∈ Ω , 0 ≤ r ≤ 2. (2)

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

A family of 3-uniform measures II

For every k = 0, 1, . . ., let Ck be the cone in Rk+4 consisting of the points x = (x1, . . . , xk+4) satisfying x ∈

• x2

4 = x2 1 + x2 2 + x2 3

• ∩

k

• l=1
• x2

l+4 = 2lx2 4

• .

Then, for all x ∈ Ck, for all r > 0 H3(B(x, r) ∩ Ck) = 4 3πr 3.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Classification in codimension 2

Let ν be a conical 3-uniform measure in R5. Then ν = cH3Σ where Σ is, up to isometry, one of the following sets: {x4 = 0} ∩ {x5 = 0} ,

x2

4 = x2 1 + x2 2 + x2 3

∩ {x5 = 0} , x2

4 = x2 1 + x2 2 + x2 3

∩ x2

5 = 2x2 4

.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Example with little symmetry

Let C be the following cone in R7: C =

• x2

4 = x2 1 + x2 2 + x2 3

• ∩
• x2

5 = 2x42

∩

• x62 + x72 = 4x2

4

• ∩ {x7p(x4, x5, x6, x7) = 0} ,

where p(x4, x5, x6, x7) = (x7 − √ 3x4)2 + (x6 − x4)2(x5 − √ 2x4)2. Then H3C is a 3-uniform measure.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Example with little symmetry

Let C be the following cone in R7: C =

• x2

4 = x2 1 + x2 2 + x2 3

• ∩
• x2

5 = 2x42

∩

• x62 + x72 = 4x2

4

• ∩ {x7p(x4, x5, x6, x7) = 0} ,

where p(x4, x5, x6, x7) = (x7 − √ 3x4)2 + (x6 − x4)2(x5 − √ 2x4)2. Then H3C is a 3-uniform measure.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

Theorem Let µ be an n-uniform measure in Rd, n ≥ 3 and Sµ be its set of

• singularities. Then:

dimH(Sµ) ≤ n − 3.

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures

Introduction Decomposition of a conical n-uniform measure First Consequence: Construction of uniform measures Second consequence: Dimension bound for the singular set

THANK YOU!

• A. Dali Nimer (University of Washington) GMT workshop at Warwick

Geometry of uniform measures