Tube formulas, uniform measures and heat content in the Heisenberg - PowerPoint PPT Presentation

Tube formulas, uniform measures and heat content in the Heisenberg group Jeremy Tyson University of Illinois at Urbana-Champaign Workshop on Geometric Measure Theory University of Warwick 13 July 2017 based on joint works with Zolt´ an Balogh & Eugenio Vecchi, Vasilis Chousionis & Valentino Magnani, Jing Wang

Introduction I will discuss recent and ongoing work, in the setting of the sub-Riemannian Heisenberg group H n , concerning the effect of intrinsic notions of curvature on the metric and analytic properties of domains and submanifolds. Specifically, a Steiner formula for the volumes of tubular Carnot–Carath´ eodory neighborhoods of submanifolds of H n , volumes of small extrinsic Kor´ anyi balls along submanifolds, with applications to the classification of uniform measures, heat content of bounded domains with smooth and noncharacteristic boundary.

Introduction 1. (Tube formulas) For A ⊂ R n and r > 0, define the r -tubular nbhd of A N ( A , r ) = { x ∈ R n : dist( x , A ) < r } . How does the volume Vol n ( N ( A , r )) depend on A for small r ?

Introduction 1. (Tube formulas) For A ⊂ R n and r > 0, define the r -tubular nbhd of A N ( A , r ) = { x ∈ R n : dist( x , A ) < r } . How does the volume Vol n ( N ( A , r )) depend on A for small r ? 2. (Volumes of extrinsic balls) Let Σ m ⊂ R n be a C k submanifold. How does Vol m ( B ( x , r ) ∩ Σ) behave for x ∈ Σ and small r ?

Introduction 1. (Tube formulas) For A ⊂ R n and r > 0, define the r -tubular nbhd of A N ( A , r ) = { x ∈ R n : dist( x , A ) < r } . How does the volume Vol n ( N ( A , r )) depend on A for small r ? 2. (Volumes of extrinsic balls) Let Σ m ⊂ R n be a C k submanifold. How does Vol m ( B ( x , r ) ∩ Σ) behave for x ∈ Σ and small r ? 3. (Heat content) Let v ( x , s ) solve the heat equation in a bounded domain U ⊂ R n with zero Dirichlet boundary conditions and initial temperature � v ( x , 0) = 1 across Ω. How does the total heat content Q ( s ) = U v ( x , s ) dx behave for small s ?

Introduction 2. (Volumes of extrinsic balls) Let Σ m ⊂ R n be a submanifold. How does Vol m ( B ( x , r ) ∩ Σ) behave for x ∈ Σ and small r ?

Introduction 2. (Volumes of extrinsic balls) Let Σ m ⊂ R n be a submanifold. How does Vol m ( B ( x , r ) ∩ Σ) behave for x ∈ Σ and small r ? Motivation comes from the study of density properties of Radon measures and their relationship to rectifiability. The s -density of a Radon measure µ at x is Θ s ( µ, x ) = lim r → 0 r − s µ ( B ( x , r )). Theorem (Besicovitch, 1938; Moore, 1950; Preiss, 1987) Let µ be a Radon measure in R n . Assume that Θ m ( µ, · ) exists in (0 , + ∞ ) µ -a.e. Then µ is m-rectifiable.

Introduction 2. (Volumes of extrinsic balls) Let Σ m ⊂ R n be a submanifold. How does Vol m ( B ( x , r ) ∩ Σ) behave for x ∈ Σ and small r ? If the m -density exists in (0 , + ∞ ) µ -a.e. in spt µ , then, for µ -a.e. a , every tangent meas ν ∈ Tan( µ, a ) is m -uniform : ∃ c > 0 s.t. ν ( B ( x , r )) = cr m ∀ x ∈ spt ν, r > 0. (Kowalski–Preiss, 1987) Let µ be an ( n − 1)-uniform measure in R n . Then µ = c H n − 1 Σ where Σ is isometric to either a hyperplane or to the light cone { x 2 1 = x 2 2 + x 2 3 + x 2 4 } . The proof, in dimensions ≥ 4, begins by deriving several geometric PDE involving curvatures of Σ from the coefficients in the power series expansion Vol n − 1 ( B ( x , r ) ∩ Σ) = Ω n − 1 r n − 1 + · · ·

The Heisenberg group H n Points in H n = R 2 n +1 denoted p = ( z , t ) = ( x , y , t ) = ( x 1 , . . . , x n , y 1 , . . . , y n , t ). ( z , t ) ∗ ( z ′ , t ′ ) = ( z + z ′ , t + t ′ + 2( x · y ′ − x ′ · y )) = ( z + z ′ , t + t ′ + 2 ω ( z , z ′ )) X 1 , . . . , X n , Y 1 = X n +1 , . . . , Y n = X 2 n , X 2 n +1 left invariant vector fields h n = v 1 ⊕ v 2 = span { X 1 , . . . , X 2 n } ⊕ span { X 2 n +1 } step two stratified Lie algebra horizontal distribution H p H n = span { V ( p ) : V ∈ v 1 } sub-Riemannian metric g 0 s.t. X 1 , . . . , X 2 n form an ON frame pseudo-Hermitian structure determined by contact form ϑ = dt + 2 � n j =1 x j dy j − y j dx j and almost complex structure J in the horizon- tal bundle such that J ( X j ) = Y j , J ( Y j ) = − X j

Metrics on the Heisenberg group H n 1. The Carnot–Carath´ eodory metric d cc ( p , q ) = inf length g 0 ( γ ), infimum over all horizontal curves γ joining p to q . 2. The gauge (or Kor´ anyi ) metric d H ( p , q ) = || p − 1 ∗ q || H || ( z , t ) || H = ( | z | 4 + t 2 ) 1 / 4 Credit: Anton Lukyanenko Remarks: (1) Dilations δ r : H n → H n , δ r ( z , t ) = ( rz , r 2 t ), commute with left translation and act as similarities of either d cc or d H . (2) d cc and d H are bi-Lipschitz equivalent. (3) Hausdorff dimension of H n is Q := 2 n + 2.

I. A Steiner tube formula for the C–C metric Fix a bounded C 2 domain U ⊂ H n . Goal: Develop a power series expansion for the function r �→ Vol( N cc ( U , r )) and identify the coefficients in terms of the induced sub-Riemannian geometry of Σ = ∂ U . Volume is the Haar measure on H n (agrees with Lebesgue measure in R 2 n +1 ).

I. A Steiner tube formula for the C–C metric Fix a bounded C 2 domain U ⊂ H n . Goal: Develop a power series expansion for the function r �→ Vol( N cc ( U , r )) and identify the coefficients in terms of the induced sub-Riemannian geometry of Σ = ∂ U . Volume is the Haar measure on H n (agrees with Lebesgue measure in R 2 n +1 ). This has been done in H 1 by Balogh–Ferrari–Franchi–Vecchi–Wildrick (2015) and in H n by Ritor´ e (2017, preprint). I’ll describe the H 1 Steiner formula and some ideas of the proof, then discuss a slight simplification via a sub-Riem Gauss–Bonnet formula (Balogh-T-Vecchi, 2016)

Volumes of tubular nbhds: example and history U ⊂ R 3 bounded, Σ = ∂ U area A , � n inward unit normal Σ t ( u , v ) = Σ( u , v ) − t � n ( u , v ) t -parallel surface d σ | Σ t = det( I + tS ) d σ = (1 + tH + t 2 K ) d σ � ∂ U H d σ ) r 2 + 1 � Vol 3 ( N ( U , r )) = Vol 3 ( U ) + rA + 1 ∂ U K d σ ) r 3 2 ( 3 ( if r < max {| k 1 | , | k 2 |} − 1

Volumes of tubular nbhds: example and history U ⊂ R 3 bounded, Σ = ∂ U area A , � n inward unit normal Σ t ( u , v ) = Σ( u , v ) − t � n ( u , v ) t -parallel surface d σ | Σ t = det( I + tS ) d σ = (1 + tH + t 2 K ) d σ � ∂ U H d σ ) r 2 + 1 � Vol 3 ( N ( U , r )) = Vol 3 ( U ) + rA + 1 ∂ U K d σ ) r 3 2 ( 3 ( if r < max {| k 1 | , | k 2 |} − 1 Steiner (1840), Weyl (1939) Definition (Federer) The reach of a closed set K is the largest r > 0 so that to each x ∈ N ( K , r ) there is a unique ξ ∈ K realizing dist( x , K ). K convex: reach( K ) = + ∞ Σ closed C 2 hypersurface: reach(Σ) = (max {| k 1 | , . . . , | k n − 1 |} ) − 1

Steiner’s tube formula: proof outline Step 1. Write � r Vol( N ( U , r )) = Vol( U ) + P ( ∂ N ( U , t ) ) dt . � �� � 0 U t � Step 2. Parameterize Φ t : ∂ U → ∂ U t , so P ( ∂ U t ) = ∂ U J Φ t dP . Step 3. Expand J Φ t in a series.

Steiner’s tube formula: proof outline Step 1. Write � r Vol( N ( U , r )) = Vol( U ) + P ( ∂ N ( U , t ) ) dt . � �� � 0 U t � Step 2. Parameterize Φ t : ∂ U → ∂ U t , so P ( ∂ U t ) = ∂ U J Φ t dP . Step 3. Expand J Φ t in a series. Step 1 relies on a coarea formula and the eikonal equation |∇ dist( · , U | = 1 a.e. For the C–C metric in sub-Riemannian spaces, the eikonal equation |∇ H dist cc ( · , U ) | = 1 is due to Monti–Serra Cassano (2001). | z | Eikonal eq’n for the Kor´ anyi metric is not true: |∇ H ( || · || H )( z , t ) | = || ( z , t ) || H ≤ 1.

How to parametrize U r \ U ? Σ ⊂ R n C 2 hypersurface, r < reach(Σ): Φ : ( − r , r ) × Σ → N (Σ , r ) , Φ( t , ξ ) = ξ + t � n ( ξ ) local parameterization Issues: (1) Initial position and velocity do not uniquely specify a geodesic in H n . (2) No positive injectivity radius. (3) Choice of normal direction?

Local structure of tubular nbhds Definition ξ ∈ Σ is a characteristic point if T ξ Σ = H ξ H n . C (Σ) = characteristic set .

Local structure of tubular nbhds Definition ξ ∈ Σ is a characteristic point if T ξ Σ = H ξ H n . C (Σ) = characteristic set . Theorem (Arcozzi–Ferrari, 2007; Ritor´ e, 2017 preprint) Compact C 2 hypersurfaces Σ without characteristic points have positive CC reach For 0 < r < reach(Σ) , p ∈ N cc (Σ , r ) and ξ ∈ Σ realizing dist cc ( p , Σ) there is a unique C–C geodesic γ : [0 , δ ] → H n joining ξ = γ (0) to p = γ ( δ ) . Initial velocity vector of γ is a multiple of the horizontal normal. Φ : ( − r , r ) × Σ → N cc (Σ , r ) , Φ( t , ξ ) = γ ( t ) local parameterization