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 R d . If Φ is such that Φ( B ( x , r )) 0 < Θ n (Φ , x ) := lim < ∞ ω n r n r → 0 for Φ-almost every x ∈ R d . Then Φ ≪ H n and Φ-almost all of R d 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 R d . If Φ is such that Φ( B ( x , r )) 0 < Θ n (Φ , x ) := lim < ∞ ω n r n r → 0 for Φ-almost every x ∈ R d . Then Φ ≪ H n and Φ-almost all of R d 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 R d . 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 R d . 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 R d are multiples of Lebesgue measure. The 1-uniform measures in R d , d ≥ 1, are multiples of H 1 � L , for some line L . The 2-uniform measures in R d , d ≥ 2, are multiples of H 2 � P , 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 R d are multiples of Lebesgue measure. The 1-uniform measures in R d , d ≥ 1, are multiples of H 1 � L , for some line L . The 2-uniform measures in R d , d ≥ 2, are multiples of H 2 � P , 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 R d are multiples of Lebesgue measure. The 1-uniform measures in R d , d ≥ 1, are multiples of H 1 � L , for some line L . The 2-uniform measures in R d , d ≥ 2, are multiples of H 2 � P , 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 ⊂ R 4 be the following set: C = { ( x 1 , x 2 , x 3 , x 4 ) ∈ R 4 ; x 2 4 = x 2 1 + x 2 2 + x 2 3 } . The measure H 3 � C 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 ⊂ R 4 be the following set: C = { ( x 1 , x 2 , x 3 , x 4 ) ∈ R 4 ; x 2 4 = x 2 1 + x 2 2 + x 2 3 } . The measure H 3 � C 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 R n +1 can only be an n -plane or (up to rotation) R n − 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 R n +1 can only be an n -plane or (up to rotation) R n − 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 R n +1 can only be an n -plane or (up to rotation) R n − 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 ⊂ R d and r > 0 ν ( rA ) = r n ν ( A ) . Reduce the study of such measures to the study of their spherical component Ω = spt ( ν ) ∩ S d − 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 ⊂ R d and r > 0 ν ( rA ) = r n ν ( A ) . Reduce the study of such measures to the study of their spherical component Ω = spt ( ν ) ∩ S d − 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 ⊂ R d and r > 0 ν ( rA ) = r n ν ( A ) . Reduce the study of such measures to the study of their spherical component Ω = spt ( ν ) ∩ S d − 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 R d . Let σ be its spherical component i.e. σ = H n − 1 � Ω where Ω = spt ( ν ) ∩ S d − 1 . Then σ is uniformly distributed: for all x ∈ Ω, for all 0 ≤ r ≤ 2 σ ( B ( x , r )) = H n − 1 � B ( e , r ) ∩ S n − 1 � , where e ∈ S n − 1 is arbitrarily chosen. A. Dali Nimer (University of Washington) GMT workshop at Warwick Geometry of uniform measures
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