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 Hn, 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 Hn, 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 ⊂ Rn and r > 0, define the r-tubular nbhd of A

N(A, r) = {x ∈ Rn : dist(x, A) < r} . How does the volume Voln(N(A, r)) depend on A for small r?

Introduction

• 1. (Tube formulas) For A ⊂ Rn and r > 0, define the r-tubular nbhd of A

N(A, r) = {x ∈ Rn : dist(x, A) < r} . How does the volume Voln(N(A, r)) depend on A for small r?

• 2. (Volumes of extrinsic balls) Let Σm ⊂ Rn be a C k submanifold. How does

Volm(B(x, r) ∩ Σ) behave for x ∈ Σ and small r?

Introduction

• 1. (Tube formulas) For A ⊂ Rn and r > 0, define the r-tubular nbhd of A

N(A, r) = {x ∈ Rn : dist(x, A) < r} . How does the volume Voln(N(A, r)) depend on A for small r?

• 2. (Volumes of extrinsic balls) Let Σm ⊂ Rn be a C k submanifold. How does

Volm(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 ⊂ Rn 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 Σm ⊂ Rn be a submanifold. How does

Volm(B(x, r) ∩ Σ) behave for x ∈ Σ and small r?

Introduction

• 2. (Volumes of extrinsic balls) Let Σm ⊂ Rn be a submanifold. How does

Volm(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 Θs(µ, x) = limr→0 r −sµ(B(x, r)). Theorem (Besicovitch, 1938; Moore, 1950; Preiss, 1987) Let µ be a Radon measure in Rn. Assume that Θm(µ, ·) exists in (0, +∞) µ-a.e. Then µ is m-rectifiable.

Introduction

• 2. (Volumes of extrinsic balls) Let Σm ⊂ Rn be a submanifold. How does

Volm(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 Rn. Then µ = c Hn−1 Σ where Σ is isometric to either a hyperplane or to the light cone {x2

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

The proof, in dimensions ≥ 4, begins by deriving several geometric PDE involving curvatures of Σ from the coefficients in the power series expansion Voln−1(B(x, r) ∩ Σ) = Ωn−1r n−1 + · · ·

The Heisenberg group Hn

Points in Hn = R2n+1 denoted p = (z, t) = (x, y, t) = (x1, . . . , xn, y1, . . . , yn, t). (z, t) ∗ (z′, t′) = (z + z′, t + t′ + 2(x · y ′ − x′ · y)) = (z + z′, t + t′ + 2ω(z, z′)) X1, . . . , Xn, Y1 = Xn+1, . . . , Yn = X2n, X2n+1 left invariant vector fields hn = v1 ⊕ v2 = span{X1, . . . , X2n} ⊕ span{X2n+1} step two stratified Lie algebra horizontal distribution HpHn = span{V (p) : V ∈ v1} sub-Riemannian metric g0 s.t. X1, . . . , X2n form an ON frame pseudo-Hermitian structure determined by contact form ϑ = dt + 2 n

j=1 xj dyj − yj dxj

and almost complex structure J in the horizon- tal bundle such that J(Xj) = Yj, J(Yj) = −Xj

Metrics on the Heisenberg group Hn

• 1. The Carnot–Carath´

eodory metric dcc(p, q) = inf lengthg0(γ), infimum over all horizontal curves γ joining p to q.

• 2. The gauge (or Kor´

anyi) metric dH(p, q) = ||p−1 ∗ q||H ||(z, t)||H = (|z|4 + t2)1/4

Credit: Anton Lukyanenko

Remarks: (1) Dilations δr : Hn → Hn, δr(z, t) = (rz, r 2t), commute with left translation and act as similarities of either dcc or dH. (2) dcc and dH are bi-Lipschitz equivalent. (3) Hausdorff dimension of Hn is Q := 2n + 2.

• I. A Steiner tube formula for the C–C metric

Fix a bounded C 2 domain U ⊂ Hn. Goal: Develop a power series expansion for the function r → Vol(Ncc(U, r)) and identify the coefficients in terms of the induced sub-Riemannian geometry of Σ = ∂U. Volume is the Haar measure on Hn (agrees with Lebesgue measure in R2n+1).

• I. A Steiner tube formula for the C–C metric

Fix a bounded C 2 domain U ⊂ Hn. Goal: Develop a power series expansion for the function r → Vol(Ncc(U, r)) and identify the coefficients in terms of the induced sub-Riemannian geometry of Σ = ∂U. Volume is the Haar measure on Hn (agrees with Lebesgue measure in R2n+1). This has been done in H1 by Balogh–Ferrari–Franchi–Vecchi–Wildrick (2015) and in Hn by Ritor´ e (2017, preprint). I’ll describe the H1 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 ⊂ R3 bounded, Σ = ∂U area A, n inward unit normal Σt(u, v) = Σ(u, v) − t n(u, v) t-parallel surface dσ|Σt = det(I + tS) dσ = (1 + tH + t2K) dσ Vol3(N(U, r)) = Vol3(U) + rA + 1

2(

• ∂U H dσ)r 2 + 1

3(

• ∂U K dσ)r 3

if r < max{|k1|, |k2|}−1

Volumes of tubular nbhds: example and history

U ⊂ R3 bounded, Σ = ∂U area A, n inward unit normal Σt(u, v) = Σ(u, v) − t n(u, v) t-parallel surface dσ|Σt = det(I + tS) dσ = (1 + tH + t2K) dσ Vol3(N(U, r)) = Vol3(U) + rA + 1

2(

• ∂U H dσ)r 2 + 1

3(

• ∂U K dσ)r 3

if r < max{|k1|, |k2|}−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{|k1|, . . . , |kn−1|})−1

Steiner’s tube formula: proof outline

Step 1. Write Vol(N(U, r)) = Vol(U) + r P(∂ N(U, t)

Ut

) dt. Step 2. Parameterize Φt : ∂U → ∂Ut, so P(∂Ut) =

• ∂U JΦt dP.

Step 3. Expand JΦt in a series.

Steiner’s tube formula: proof outline

Step 1. Write Vol(N(U, r)) = Vol(U) + r P(∂ N(U, t)

Ut

) dt. Step 2. Parameterize Φt : ∂U → ∂Ut, so P(∂Ut) =

• ∂U JΦt dP.

Step 3. Expand JΦt 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 distcc(·, U)| = 1 is due to Monti–Serra Cassano (2001). Eikonal eq’n for the Kor´ anyi metric is not true: |∇H(|| · ||H)(z, t)| =

|z| ||(z,t)||H ≤ 1.

How to parametrize Ur \ U?

Σ ⊂ Rn 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 Hn. (2) No positive injectivity radius. (3) Choice of normal direction?

Local structure of tubular nbhds

Definition ξ ∈ Σ is a characteristic point if TξΣ = HξHn. C(Σ) = characteristic set.

Local structure of tubular nbhds

Definition ξ ∈ Σ is a characteristic point if TξΣ = HξHn. 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 ∈ Ncc(Σ, r) and ξ ∈ Σ realizing distcc(p, Σ) there is a unique C–C geodesic γ : [0, δ] → Hn joining ξ = γ(0) to p = γ(δ). Initial velocity vector of γ is a multiple of the horizontal normal. Φ : (−r, r) × Σ → Ncc(Σ, r), Φ(t, ξ) = γ(t) local parameterization

Local structure of tubular nbhds

Definition ξ ∈ Σ is a characteristic point if TξΣ = HξHn. 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 ∈ Ncc(Σ, r) and ξ ∈ Σ realizing distcc(p, Σ) there is a unique C–C geodesic γ : [0, δ] → Hn joining ξ = γ(0) to p = γ(δ). Initial velocity vector of γ is a multiple of the horizontal normal. Φ : (−r, r) × Σ → Ncc(Σ, r), Φ(t, ξ) = γ(t) local parameterization Remark: The noncharacteristic assumption is somewhat restrictive. For instance, if Σ is a topological sphere then C(Σ) = ∅. There are localized versions of the discussion which allow one to work in compact regions of Σ \ C(Σ).

Geodesics in Hn

C–C geodesics starting at p ∈ Hn are parameterized by v ∈ S2n−1 and a curvature parameter λ ∈ R. γλ

p, v(s) = p ∗ (x1(s), . . . , xn(s), y1(s), . . . , yn(s), t(s))

xi(s) = vi sin(λs) λ

• + vn+i

1 − cos(λs) λ

• yi(s) = vn+i

sin(λs) λ

• − vi

1 − cos(λs) λ

• if λ = 0, or

xi(s) = vi s yi(s) = vn+i s if λ = 0. Vertical component t(s) is determined by the horizontality condition t′(s) + 2 n

j=1(xj(s)x′ n+j(s) − xn+j(s)x′ j (s)) = 0.

Sub-Riemannian differential geometry of submflds of Hn

Let g1 be the Riemannian metric on Hn s.t. X1, . . . , X2n+1 form an ON frame. For a C 2 hypersurface Σ = ∂U ⊂ Hn:

• n1 g1 outer unit normal,
• n0 projection of

n1 to HHn,

• ν0 =

n0/| n0|1 unit outer horizontal normal (defined in Σ \ C(Σ))

• e1 = J(

ν0) characteristic direction (spans HΣ = TΣ ∩ HH1) H0 = div0( ν0) horizontal mean curvature

Sub-Riemannian differential geometry of submflds of Hn

Let g1 be the Riemannian metric on Hn s.t. X1, . . . , X2n+1 form an ON frame. For a C 2 hypersurface Σ = ∂U ⊂ Hn:

• n1 g1 outer unit normal,
• n0 projection of

n1 to HHn,

• ν0 =

n0/| n0|1 unit outer horizontal normal (defined in Σ \ C(Σ))

• e1 = J(

ν0) characteristic direction (spans HΣ = TΣ ∩ HH1) H0 = div0( ν0) horizontal mean curvature For p ∈ Ncc(Σ, r) with closest point ξ ∈ Σ, the connecting geodesic γλ

ξ, v has

initial velocity vector v = ν0(ξ) and curvature λ = 4 n1, X2n+11 | n0|1 (ξ) =: P0(ξ) (‘imaginary curvature’)

Steiner tube formula in H1

(Balogh–Ferrari–Franchi–Vecchi–Wildrick, 2015; Ritor´ e, 2017 preprint) Σ = ∂U C 2 noncharacteristic, U ⊂ H1 bounded domain Vol(Ncc(U, r)) = Vol(U) + σ0(Σ)r + 1 2

• Σ

H0 dσ0 r 2 − 1 6

• Σ
• P2

0 + 2

e1(P0)

• dσ0 r 3 + o(r 3)

dσ0 = | n0| dσ1 horizontal perimeter measure on Σ

Steiner tube formula in H1

(Balogh–Ferrari–Franchi–Vecchi–Wildrick, 2015; Ritor´ e, 2017 preprint) Σ = ∂U C 2 noncharacteristic, U ⊂ H1 bounded domain Vol(Ncc(U, r)) = Vol(U) + σ0(Σ)r + 1 2

• Σ

H0 dσ0 r 2 − 1 6

• Σ
• P2

0 + 2

e1(P0)

• dσ0 r 3 + o(r 3)

dσ0 = | n0| dσ1 horizontal perimeter measure on Σ Remark: Further terms can be computed. The power series expansion is not polynomial, but the successive integrands satisfy finite order recursions, so the full expression for Vol(Ncc(U, r)) can be summed explicitly (Ritor´ e): Vol(Ncc(U, r)) = Vol(U) +

4

• i=0
• Σ

r ai gi(P0 s)si ds

• dσ0

where a0 = 1, a1 = H0, a2 = −2 e1(P0), . . . and g0(u) = cos(u), g1(u) = sin(u)/u, g2(u) = (1 − cos(u))/u2, . . .

Steiner tube formula in H1

(Balogh–Ferrari–Franchi–Vecchi–Wildrick, 2015; Ritor´ e, 2017 preprint) Σ = ∂U C 2 noncharacteristic, U ⊂ H1 bounded domain Vol(Ncc(U, r)) = Vol(U) + σ0(Σ)r + 1 2

• Σ

H0 dσ0 r 2 − 1 6

• Σ
• P2

0 + 2

e1(P0)

• dσ0 r 3 + o(r 3)

dσ0 = | n0| dσ1 horizontal perimeter measure on Σ Remark: The expression K0 := −P2

0 −

e1(P0) is a type of intrinsic Gauss

• curvature. Using a Riemannian approximation scheme and appealing to

Gauss–Bonnet for surfaces embedded in 3-manifolds, we can prove a sub-Riem Gauss–Bonnet theorem

• Σ K0 dσ0 = 0 (Balogh–T–Vecchi, 2017), which

simplifies the above expansion: Vol(Ncc(U, r)) = Vol(U) + σ0(Σ)r + 1 2

• Σ

H0 dσ0 r 2 + 1 6

• Σ

P2

0 dσ0 r 3 + o(r 3)

Steiner tube formula in H1

(Balogh–Ferrari–Franchi–Vecchi–Wildrick, 2015; Ritor´ e, 2017 preprint) Σ = ∂U C 2 noncharacteristic, U ⊂ H1 bounded domain Vol(Ncc(U, r)) = Vol(U) + σ0(Σ)r + 1 2

• Σ

H0 dσ0 r 2 − 1 6

• Σ
• P2

0 + 2

e1(P0)

• dσ0 r 3 + o(r 3)

dσ0 = | n0| dσ1 horizontal perimeter measure on Σ Question: Let (X, d, µ) be doubling with 1-PI, U ⊂ X a bounded domain with finite perimeter. Assume that ∂U has positive mean curvature in the sense of L–M–Sh–Sp. Under what conditions do we have lim inf

r→0

µ(Nd(U, r) \ U) − P(U)r r 2 > 0 ?

• II. Uniform measures and density theorems in Hn

Recall: The s-density of a Radon measure µ in a metric space (X, d) is Θs(µ, x) := lim

r→0 r −sµ(B(x, r)).

Theorem (Besicovitch, 1938; Moore, 1950; Preiss, 1987) Let µ be a Radon measure in Rn. Assume that Θm(µ, ·) exists in (0, +∞) µ-a.e. Then µ is m-rectifiable. Theorem (Marstrand, 1964) Let µ be a Radon measure in Rn such that Θs(µ, ·) exists and is positive and finite in a set of positive µ measure. Then s is an integer.

• II. Uniform measures and density theorems in Hn

As a first step towards understanding the relationship between densities and rectifiability in the sub-Riemannian setting, we extended Marstrand’s theorem. Theorem (Chousionis–T, 2015) Let µ be a Radon measure in (Hn, dH) s.t. the density Θs(µ, ·) exists and is positive and finite in a set of positive µ measure. Then s is an integer. Remarks: 1. s may not agree with the topological dimension of spt(µ).

• 2. The support of µ need not be rectifiable. For instance, if µ is Hausdorff

2-measure restricted to the vertical axis T := {(0, t) : t ∈ R}, then Θ2(µ, p) is a positive constant for all p ∈ T. But T is not a rectifiable curve in Hn.

• 3. Marstrand’s theorem for (Hn, dcc) is not known.

Marstrand’s Density Theorem in (Hn, dH)

Two proofs in the literature for Marstrand’s theorem dimension reduction and blow-up argument, (Kirchheim–Preiss): analyze the supports of uniform and uniformly distributed measures. ν uniformly distributed if there exists a function f : (0, ∞) → R s.t. ν(B(x, r)) = f (r) for all x ∈ spt ν. Theorem (Kirchheim–Preiss, 2002) Let ν be a uniformly distributed measure in Rn. Then spt(ν) is an analytic variety.

Marstrand’s Density Theorem in (Hn, dH)

Theorem (Chousionis–T) Let ν be a uniformly distributed measure in (Hn, dH). Then spt(ν) is an analytic variety (in R2n+1). For fixed p0 ∈ spt(ν), spt(ν) is the common zeroset of the real analytic functions Fs(p) =

• Hn
• exp(−s dH(p, q)4) − exp(−s dH(p0, q)4)
• dν(q),

s > 0.

Homogeneous subgroups and dimensions of submanifolds

A subgroup G in Hn is homogeneous if it is closed under dilations. horizontal: G = V = V × {0}, V an isotropic subspace of (R2n, ω) These exist only in dimensions 1 ≤ m ≤ n. vertical: G = W = W × R, W any proper subspace of R2n Intrinsic geometry of horizontal subspaces is Euclidean: dimH V = dim V . Vertical subspaces exhibit a dimension jump: dimH W = dim W + 1 = dim W + 2. Intrinsic geometry is not Euclidean. (Gromov) Σ ⊂ Hn m-dimensional C 1,1 submfld ⇒ dimH Σ = m ∈ {m, m + 1}.

Marstrand theorem: overview of the proof

Θs(µ, ·) exists in (0, +∞) on a set of positive µ measure ∃ an s-uniform measure ν spt(ν) is a real analytic variety spt(ν) is a union of real analytic submanifolds dimH spt(ν) ∈ N s ∈ N (Kirchheim–Preiss theorem) (Lojasiewicz Structure Theorem) (Gromov’s Theorem)

A comment about rectifiability in Hn

A comment about rectifiability in Hn

According to a standard definition of rectifiability in Hn (Franchi–Serapioni–Serra Cassano), the countably (m, H)-rectifiable subsets of Hn are those sets which are covered, up to a set of Hm

H measure zero, by a countable family of H-regular

m-dimensional submanifolds. Here m = m if 1 ≤ m ≤ n, and m = m − 1 if n + 1 ≤ m ≤ 2n. H-regular submflds are defined as C 1 images of subsets of Rm (if 1 ≤ m ≤ n), and as level sets of C 1

H maps f : Hn → R2n+1−m (if n + 1 ≤ m ≤ 2n).

But Sm

H

G is m-uniform for any homogeneous subgroup G, including ‘non-rectifiable’ pairs (m, m + 1) for m = 1, . . . , n.

Uniform measures in (H1, dH)

Conjecture All uniform measures in H1 are flat. More precisely, let ν be an m-uniform measure in (H1, dH). Then spt ν is a left coset of a homogeneous subgroup of metric dimension m and topological dimension m, and ν = c Sm

H

spt ν

Uniform measures in (H1, dH)

Conjecture All uniform measures in H1 are flat. More precisely, let ν be an m-uniform measure in (H1, dH). Then spt ν is a left coset of a homogeneous subgroup of metric dimension m and topological dimension m, and ν = c Sm

H

spt ν (ν is (m, m)-flat).

Uniform measures in (H1, dH)

Conjecture All uniform measures in H1 are flat. More precisely, let ν be an m-uniform measure in (H1, dH). Then spt ν is a left coset of a homogeneous subgroup of metric dimension m and topological dimension m, and ν = c Sm

H

spt ν (ν is (m, m)-flat). Theorem (Chousionis–Magnani–T, in preparation) (A) Every (locally) 1-uniform measure in (H, dH) is (1, 1)-flat. (B) Let ν be a (locally) 2-uniform measure in (H, dH) and assume that spt ν is contained in an affine vertical plane. Then ν is (1, 2)-flat. (C) Let ν be a (locally) 3-uniform measure in (H, dH) and assume that spt ν is contained in a vertically ruled surface. Then ν is (2, 3)-flat.

Volumes of small extrinsic balls on submanifolds of H

The previous theorem is a consequence of explicit power series formulas for the volumes of small extrinsic balls in the Kor´ anyi metric along submanifolds. Sm(BdH(p, r) ∩ Σ) = c0 r m + · · · The value of Sm

H (BdH(p, r) ∩ Σ) is computed using a sub-Riemannnian area

formula (Magnani).

Riemannian approximation scheme

Let gǫ be the Riemannian metric on R3 s.t. X1, X2 and ǫ1/2X3 are an ON frame for TH. Philosophy: Analyse the sequence (R3, gǫ) as ǫ → 0. What differential geometric information survives in the limit? Remarks: (1) (R3, gǫ) converges (in the pointed Gromov–Hausdorff sense) to (H, g0) (Gromov). (2) The Ricci curvatures of (R3, gǫ) diverge to both +∞ and −∞ as ǫ → 0.

Curvature of curves

Let Σ = (x, y, t) = (γ, t) be a C 2 smooth curve, kǫ

Σ its curvature in (R3, gǫ).

kǫ

Σ ǫ→0

− → k0

Σ(s) :=

| ˙

x ¨ y− ˙ y ¨ x| | ˙ γ|3

, at interior horizontal points Σ(s),

| ˙ γ| ϑ( ˙ Σ),

at nonhorizontal points Σ(s). Recall: ϑ = dt + 2x dy − 2y dx contact form in H1 Σ horiz at p = Σ(s) if ϑ( ˙ Σ)(s) = (˙ γ3 + 2γ1 ˙ γ2 − 2γ2 ˙ γ1)(s) = 0

| ˙ x ¨ y− ˙ y ¨ x| | ˙ γ|3

= κγ = (unsigned) curvature of a plane curve γ = (x, y)

Volumes of extrinsic balls along curves

• 1. Let Σ = (γ, t) be a C 2 horizontal curve in H1.

S1

H(BH(p, r) ∩ Σ) = 2r + a1(p)r 3 + · · · ,

a1(p) =

1 36 k0 Σ

Volumes of extrinsic balls along curves

• 1. Let Σ = (γ, t) be a C 2 horizontal curve in H1.

S1

H(BH(p, r) ∩ Σ) = 2r + a1(p)r 3 + · · · ,

a1(p) =

1 36 k0 Σ

S1

H

Σ 1-uniform ⇒ k0

Σ = 0 ⇒ γ is a line in R2 ⇒ Σ is a horizontal line in H.

Volumes of extrinsic balls along curves

• 1. Let Σ = (γ, t) be a C 2 horizontal curve in H1.

S1

H(BH(p, r) ∩ Σ) = 2r + a1(p)r 3 + · · · ,

a1(p) =

1 36 k0 Σ

S1

H

Σ 1-uniform ⇒ k0

Σ = 0 ⇒ γ is a line in R2 ⇒ Σ is a horizontal line in H.

• 2. Let Σ = (γ, t) be a C 2 nonhorizontal curve in H1, assume ϑ( ˙

Σ) = 1. S2

H(BH(p, r) ∩ Σ) = 2r 2 + b1(p)r 6 + · · ·

b1(z, t) = 0 iff 2

3κγ(s) = |˙

γ(s)|.

Volumes of extrinsic balls along curves

• 1. Let Σ = (γ, t) be a C 2 horizontal curve in H1.

S1

H(BH(p, r) ∩ Σ) = 2r + a1(p)r 3 + · · · ,

a1(p) =

1 36 k0 Σ

S1

H

Σ 1-uniform ⇒ k0

Σ = 0 ⇒ γ is a line in R2 ⇒ Σ is a horizontal line in H.

• 2. Let Σ = (γ, t) be a C 2 nonhorizontal curve in H1, assume ϑ( ˙

Σ) = 1. S2

H(BH(p, r) ∩ Σ) = 2r 2 + b1(p)r 6 + · · ·

b1(z, t) = 0 iff 2

3κγ(s) = |˙

γ(s)|. S2

H

Σ 2-uniform and Σ contained in a vertical plane ⇒ κγ = 0 ⇒ ˙ γ = 0 ⇒ Σ is a vertical line.

Volumes of extrinsic balls along curves

• 1. Let Σ = (γ, t) be a C 2 horizontal curve in H1.

S1

H(BH(p, r) ∩ Σ) = 2r + a1(p)r 3 + · · · ,

a1(p) =

1 36 k0 Σ

S1

H

Σ 1-uniform ⇒ k0

Σ = 0 ⇒ γ is a line in R2 ⇒ Σ is a horizontal line in H.

• 2. Let Σ = (γ, t) be a C 2 nonhorizontal curve in H1, assume ϑ( ˙

Σ) = 1. S2

H(BH(p, r) ∩ Σ) = 2r 2 + b1(p)r 6 + · · ·

b1(z, t) = 0 iff 2

3κγ(s) = |˙

γ(s)|. S2

H

Σ 2-uniform and Σ contained in a vertical plane ⇒ κγ = 0 ⇒ ˙ γ = 0 ⇒ Σ is a vertical line. Remark: Without the assumption ϑ( ˙ Σ) = 1 we have b1(z, t) = 0 iff

2 3κγ(s) = | ˙ γ(s)| ϑ( ˙ Σ)(s).

Curvatures and volumes of extrinsic balls along surfaces

• 3. Σ C 2 smooth surface, p ∈ Σ noncharacteristic

H0 = div0( n0/| n0|1) horizontal mean curvature P0 = 4 n1, X31/| n0|1 Arcozzi–Ferrari imaginary curvature The series expansion for the spherical Hausdorff 3-measure of small balls along Σ is S3

H(BH(p, r) ∩ Σ) = c0 r 3 + c1(p)r 5 + · · ·

where c1(p) = c1 · (H0(p)2 + 3

2P0(p)2 + 4

e1(P0)).

Curvatures and volumes of extrinsic balls along surfaces

• 3. Σ C 2 smooth surface, p ∈ Σ noncharacteristic

H0 = div0( n0/| n0|1) horizontal mean curvature P0 = 4 n1, X31/| n0|1 Arcozzi–Ferrari imaginary curvature The series expansion for the spherical Hausdorff 3-measure of small balls along Σ is S3

H(BH(p, r) ∩ Σ) = c0 r 3 + c1(p)r 5 + · · ·

where c1(p) = c1 · (H0(p)2 + 3

2P0(p)2 + 4

e1(P0)). S3

H

Σ 3-uniform ⇒ all points are noncharacteristic and the PDE (1) H0(p)2 + 3

2P0(p)2 + 4

e1(P0) ≡ 0 is satisfied everywhere along Σ.

Curvatures and volumes of extrinsic balls along surfaces

• 3. Σ C 2 smooth surface, p ∈ Σ noncharacteristic

H0 = div0( n0/| n0|1) horizontal mean curvature P0 = 4 n1, X31/| n0|1 Arcozzi–Ferrari imaginary curvature The series expansion for the spherical Hausdorff 3-measure of small balls along Σ is S3

H(BH(p, r) ∩ Σ) = c0 r 3 + c1(p)r 5 + · · ·

where c1(p) = c1 · (H0(p)2 + 3

2P0(p)2 + 4

e1(P0)). S3

H

Σ 3-uniform ⇒ all points are noncharacteristic and the PDE (1) H0(p)2 + 3

2P0(p)2 + 4

e1(P0) ≡ 0 is satisfied everywhere along Σ. If in addition Σ is vertically ruled, then P0 = 0, so (1) reads H0 = 0, i.e., Σ is a complete noncharacteristic H-minimal vertically ruled surface. The only such

• bjects are vertical planes.

Curvatures of surfaces and a sub-Riem Gauss–Bonnet thm

Complete { νǫ, e1} to an ON frame { νǫ, e1, e2} for the tangent bundle of (Σ, gǫ). Let II ǫ be the second fundamental form for the embedding Σ ֒ → (R3, gǫ).

Curvatures of surfaces and a sub-Riem Gauss–Bonnet thm

Complete { νǫ, e1} to an ON frame { νǫ, e1, e2} for the tangent bundle of (Σ, gǫ). Let II ǫ be the second fundamental form for the embedding Σ ֒ → (R3, gǫ). Hǫ := tr II ǫ converges to the horizontal mean curvature H0.

Curvatures of surfaces and a sub-Riem Gauss–Bonnet thm

Complete { νǫ, e1} to an ON frame { νǫ, e1, e2} for the tangent bundle of (Σ, gǫ). Let II ǫ be the second fundamental form for the embedding Σ ֒ → (R3, gǫ). Hǫ := tr II ǫ converges to the horizontal mean curvature H0. det II ǫ does not converge. However, Gauss’ equation Kǫ = Kǫ + det II ǫ implies that Kǫ → K0 = −P2

0 −

e1(P0), the intrinsic Gauss curvature.

Curvatures of surfaces and a sub-Riem Gauss–Bonnet thm

Complete { νǫ, e1} to an ON frame { νǫ, e1, e2} for the tangent bundle of (Σ, gǫ). Let II ǫ be the second fundamental form for the embedding Σ ֒ → (R3, gǫ). Hǫ := tr II ǫ converges to the horizontal mean curvature H0. det II ǫ does not converge. However, Gauss’ equation Kǫ = Kǫ + det II ǫ implies that Kǫ → K0 = −P2

0 −

e1(P0), the intrinsic Gauss curvature. The horizontal perimeter measure dσ0 is the limit of rescaled Riemannian perimeter measures √ǫ dσǫ. Passing to the limit in the classical Gauss–Bonnet theorem for regular surfaces in the Riem mflds (R3, gǫ) we obtain Theorem (Balogh–T–Vecchi, 2017) Let Σ ⊂ H1 be a closed, oriented C 2 surface. Assume that H1

E(C(Σ)) = 0 and

| n0|−1

1

∈ L1(Σ, dσ1). Then

• Σ K0 dσ0 = 0.

• III. Heat content

Fix U ⊂ Rn bounded. Let v(x, s) solve the heat eq’n vs = △v in U with Dirichlet zero boundary conditions and initial temperature v|U×{0} = 1. The heat content

• f U at time s is QU(s) :=
• U v(x, s) dx.

• III. Heat content

Fix U ⊂ Rn bounded. Let v(x, s) solve the heat eq’n vs = △v in U with Dirichlet zero boundary conditions and initial temperature v|U×{0} = 1. The heat content

• f U at time s is QU(s) :=
• U v(x, s) dx.

van den Berg–Le Gall (1994): for a bounded domain U ⊂ Rn with C 3 boundary Σ, QU(s) = Vol(U) −

• 4s

π σ(Σ) + s 2

• Σ

H dσ + o(s) Theorem (T–Wang, 2017 preprint) Let U ⊂ H1 be a C 3 domain with noncharacteristic boundary Σ = ∂U. Let v(p, s) solve vs = △0v in U × (0, ∞), where △0 = X 2 + Y 2, with Dirichlet zero boundary conditions and initial temperature v(p, 0) ≡ 1. Then

• U

v(p, s) d Vol(p) = Vol(U) −

• 4s

π σ0(Σ) + s 2

• Σ

H0 dσ0 + o(s).

Heat content

• Remarks. (1) Noncharacteristic boundary is assumed in order to appeal to the

work of Arcozzi–Ferrari / Ritor´ e on the structure of tubular C–C nbhds. We need a uniform lower bound on the reach of Σ. Question: What is the situation for domains with characteristic boundary points? H0 is not defined at characteristic points, however, ||H0||L1(Σ,σ0) makes sense regardless (Danielli–Garofalo–Nhieu).

Heat content

• Remarks. (1) Noncharacteristic boundary is assumed in order to appeal to the

work of Arcozzi–Ferrari / Ritor´ e on the structure of tubular C–C nbhds. We need a uniform lower bound on the reach of Σ. Question: What is the situation for domains with characteristic boundary points? H0 is not defined at characteristic points, however, ||H0||L1(Σ,σ0) makes sense regardless (Danielli–Garofalo–Nhieu). (2) Proof is probabilistic; write v(p, s) as an exit time probability for Brownian motion starting at p. It suffices to consider small tubular nbhds of ∂U; we work in a local coordinate system defined by the A–F/Ritor´ e structure theory. After several reductions we are left to analyze the standard stochastic process (B1

s , B2 s , As), where As =

s

0 (B1 σ dB2 σ − B2 σ dB1 σ) is L´

evy’s area process.