Remarks on Hausdorff Measure and Differentiability

On Hausdorff measure and differentiability

Valentino Magnani

Remarks on Hausdorff measure and differentiability Area formulae for Lipschitz Warwick Mathematics Institute mappings The Fractal Conference on Geometric Measure Theory Geometry inside homogeneous groups

Measure Valentino Magnani theoretic area formula

Applications to University of Pisa submanifolds

July 11, 2017 On Hausdorff measure and differentiability Area formulae for Lipschitz mappings

On Hausdorff Area formulae to compute the Hausdorff measure of sets measure and can be obtained in the general context of metric spaces. differentiability Valentino [Bernd Kirchheim, Proc. Amer. Math. Soc., 1994] Magnani Consider an injective Lipschitz map f : A → X from Area formulae for k Lipschitz A ⊂ R to a metric space X. Then there holds mappings The Fractal Z Geometry inside Hk (f (A)) = Jf (x)dx. homogeneous groups A Measure k theoretic area For a.e. x ∈ A there exists a seminorm s : R → [0, +∞) formula such that Applications to submanifolds d(f (y), f (x)) − s(y − x) = o(|y − x|) as y → x, the metric differential. Its metric Jacobian is Ln x ∈ Rk : |x| ≤ 1 Jf (x) = Js = . Hk x ∈ Rk : s(x) ≤ 1 On Hausdorff measure and differentiability Area formulae for Lipschitz mappings

On Hausdorff measure and [B. Kirchheim, V.M., Proc. Edinb. Math. Soc. 2003] differentiability There exist a Lipschitz map f : → X that is nowhere Valentino H Magnani metrically differentiable, where H is the ﬁrst Heisenberg Area formulae for group. Lipschitz mappings Here metric differentiability is adapted to the group The Fractal Geometry inside operation: homogeneous groups

Measure d(f (x), f (xh)) − s(h) = o(dH(h, 0)) as h → 0, theoretic area formula

Applications to where s : H → [0, +∞) satisﬁes submanifolds 1 s(xy) ≤ s(x) + s(y) for x, y ∈ H, 2 s(δr x) = rs(x) for every x ∈ H and r > 0. This homogeneous seminorm deﬁnes the metric differential when the source space is the Heisenberg group or a more general nilpotent homogeneous group. On Hausdorff measure and differentiability Area formulae for Lipschitz mappings

On Hausdorff measure and differentiability We can see a homogeneous Lie group G as Rn equipped Valentino with a polynomial operation and a suitable left invariant Magnani distance d : G × G → [0, +∞). Area formulae for Lipschitz mappings

[V. M., T. Rajala, IMRN 2014] The Fractal Geometry inside Let A ⊂ G and let Q be the Hausdorff dimension of G. If homogeneous groups f : A → X is Lipschitz and a.e. metrically differentiable, Measure then theoretic area Z formula Q H (f (A)) = Jf (x)dx Applications to A submanifolds ( ) HQ ( )/HQ( ) where the metric Jacobian Jf x at x is sx B1 d B1 and sx is the metric differential of f at x. However, in this theorem we were forced to assume the a.e. metric differentiability, since in general it does hold. On Hausdorff measure and differentiability The Fractal Geometry inside homogeneous groups

On Hausdorff Locally on a compact set K ⊂ G we have measure and differentiability −1 1/ι C |x − y| ≤ d(x, y) ≤ C |x − y| (1) Valentino Magnani

whenever x, y ∈ K and C > 0 depends on K . Area formulae for Lipschitz mappings The estimates (1) have sharp exponents, then showing The Fractal how the Hausdorff dimension of subsets of with respect Geometry inside G homogeneous to d can be in general greater than their topological groups dimension. This is a typical feature of fractals. Measure theoretic area formula

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✄ ① On Hausdorff measure and differentiability The Fractal Geometry inside homogeneous groups

On Hausdorff One can look more carefully to the estimates on the measure and distance d. Indeed, the group G is anisotropic also at differentiability Valentino inﬁnitesimal scales. Magnani

A homogeneous group G can be broadly regarded as a Area formulae for Lipschitz suitable direct sum of subspaces mappings

The Fractal 1 ι Geometry inside G = H ⊕ · · · ⊕ H homogeneous groups along with some intrinsic dilations δ : → such that Measure r G G theoretic area formula j δr |Hj (x) = r x for j = 1, . . . , ι. Applications to submanifolds The homogeneous distance d : G × G → [0, +∞) is compatible with both group operation and dilations

d(xz, xw) = d(z, w) and d(δr z, δr w) = r d(z, w)

for each x, z, w ∈ G and r > 0. On Hausdorff measure and differentiability The Fractal Geometry inside homogeneous groups

On Hausdorff The properties of the distance make explicit the measure and “inﬁnitesimal anisotropy” of the group: differentiability −1 −1 Valentino C |x − y| ≤ d(x, y) ≤ C |x − y| if x y ∈ H1 Magnani −1p p −1 C |x − y| ≤ d(x, y) ≤ C |x − y| if x y ∈ H Area formulae for 2 Lipschitz mappings ...... The Fractal Geometry inside −1 1/ι 1/ι −1 homogeneous C |x − y| ≤ d(x, y) ≤ C |x − y| if x y ∈ Hι groups whenever x, y ∈ K and C > 0 depends on K . Measure theoretic area formula

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● ① On Hausdorff measure and differentiability The Fractal Geometry inside homogeneous groups

On Hausdorff measure and differentiability

Valentino

Magnani

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❥ ✕ ✁ ❥ ✔ Area formulae for Lipschitz

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✕ ✖ ✗ The Fractal Geometry inside ② homogeneous

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✕ ✖ ✘ theoretic area formula

Applications to −1 Along directions of degree two, namely x y ∈ H2, we have submanifolds

d(x, y)2 = d(x −1y, 0)2 ≈ |x − y|.

For a smooth curve moving along these directions we get its locally ﬁnite and positive 2-dimensional Hausdorff measure with respect to d. On Hausdorff measure and differentiability The Fractal Geometry inside homogeneous groups

On Hausdorff measure and differentiability

Valentino

Magnani

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✝ ✛ ✣ On Hausdorff measure and differentiability The Fractal Geometry inside homogeneous groups

On Hausdorff This inﬁnitesimal anisotropy implies for instance in the measure and ﬁrst Heisenberg group H ≈ (R3, xy = p(x, y), d) that differentiability Valentino 0 < H3(Σ) < +∞, Magnani

where Σ ⊂ is any 2-dimensional smooth and compact Area formulae for H Lipschitz submanifold. mappings The Fractal [L. Ambrosio, B. Kirchheim, Math. Ann. 2000] Geometry inside homogeneous The ﬁrst Heisenberg group H is purely k-unrectiﬁable for groups Measure every k ≥ 2. theoretic area formula

Applications to Comments submanifolds 1 This unrectiﬁability result shows that there are no rectiﬁable metric k-currents in H with k ≥ 2. 2 Finding a working theory of currents in Heisenberg groups or more general homogeneous groups requires working notion of “area”. On Hausdorff measure and differentiability Measure theoretic area formula

On Hausdorff Computing the Hausdorff measure of surfacesmeasure and differentiability

Valentino In particular also the 2-dimensional smooth submanifold Magnani Σ ⊂ cannot be k-rectifiable for k ≥ 2. Clearly Σ cannot H Area formulae for be of countably 1-rectifiable (otherwise Σ would be Lipschitz mappings σ-finite with respect to the one dimensional Hausdorff The Fractal measure). Geometry inside homogeneous groups

3 Measure The formula for H (Σ) is not yet known. theoretic area formula

Applications to Since Σ is not rectiﬁable with respect to the distance d of submanifolds the Heisenberg group, the standard methods to compute H3(Σ) do not apply.

The formula for H3(Σ) is a work in progress. We will focus our attention on computing the spherical measure S3(Σ). On Hausdorff measure and differentiability Measure theoretic area formula

On Hausdorff measure and [V. M., Proc. Royal. Soc. Ed., 2015] differentiability

If Σ ⊂ X is a Borel set, the diameter of balls is continuous, Valentino µ is ﬁnite on each open set of an open countable cover of Magnani α X, µ is supported on Σ and µ << S Σ, then Area formulae for x Lipschitz mappings α α µ = θ (µ, ·) S Σ. (2) The Fractal x Geometry inside homogeneous The key of this formula is the “formally explict” groups θα Measure representation of the density . theoretic area This is the spherical Federer α-density: formula Applications to submanifolds α µ(B) θ (µ, x) = inf sup α : B ∈ Fb, x ∈ B, diamB ≤ ε , ε>0 cα(2r) that is Sα measurable.

Notice that the balls B are deﬁned with respect to the distance d. On Hausdorff measure and differentiability Measure theoretic area formula

On Hausdorff The general method measure and differentiability

Valentino 1 We ﬁx an auxiliary measure µ on Σ such that Magnani µ << Sα Σ. For instance the Riemannian or the x Area formulae for Σ Lipschitz Euclidean Hausdorff measure of work well in mappings

general homogeneous groups. The Fractal Geometry inside 2 We deﬁne a µ-measurable functions Θµ,α such that homogeneous groups

α ω(d, α) Measure θ (µ, ·) = theoretic area Θµ,α formula Applications to where ω(d, α) is chosen in order to play the role of a submanifolds geometric constant. 3 The measure theoretic area formula yields Z α α Sd (Σ) := ω(d, α)S (Σ) = Θµ,α(x) dµ(x). Σ On Hausdorff measure and differentiability Applications to submanifolds

On Hausdorff Applications to submanifolds measure and differentiability

Valentino It is possible to ﬁnd the “right measure” on Σ such that Magnani

ΘµΣ,N ≡ 1. Area formulae for Lipschitz This is the natural intrinsic measure on Σ. mappings The Fractal Geometry inside [V. M. and D. Vittone, J. Reine Ang. Math., 2008] homogeneous groups If τΣ is a unit tangent n-vector with respect to a Measure Riemannian metric g and N is the Hausdorff dimension of theoretic area Σ, then the intrinsic measure of Σ is given by formula Applications to submanifolds µ = |π (τ )| Hn Σ, Σ N Σ dg x From the general procedure, if we are able to compute the Federer density of µΣ, we get N S = µΣ. On Hausdorff measure and differentiability Applications to submanifolds

On Hausdorff Regular and singular points measure and differentiability Computing the spherical Federer density of Σ at x Valentino Magnani requires to understand the blow-up limit Area formulae for −1 + Lipschitz δ1/r x Σ as r → 0 . (3) mappings The Fractal Geometry inside It is an anisotropic limit! homogeneous groups

Measure theoretic area formula

Applications to

submanifolds

✝ ① On Hausdorff measure and differentiability Applications to submanifolds

On Hausdorff measure and differentiability

Valentino Magnani

Area formulae for Lipschitz

mappings

❃ ✵ ♠❛❧❧ The Fractal Geometry inside homogeneous groups

Measure theoretic area formula

Applications to

submanifolds

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✍ ✭ ① ✝✮ ✶ ❂ On Hausdorff measure and differentiability Applications to submanifolds

On Hausdorff measure and differentiability

Valentino Magnani

Area formulae for Lipschitz

mappings

❃ ✵ ♠❛❧❧ ❡ The Fractal Geometry inside homogeneous groups

Measure theoretic area formula

Applications to

submanifolds

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✍ ✭ ① ✝✮ ✶ ❂ ✁ On Hausdorff measure and differentiability Applications to submanifolds

On Hausdorff measure and differentiability

Valentino Magnani

Area formulae for Lipschitz mappings

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Measure theoretic area formula

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❆ ✝ ✐ ❤❡ ✁♦♠♦❣✂♥ ✂♦✉✄ ☎✆♥ ❣✂♥ ☎ ✄ ♣✆❝✂ ✞ ❢ ✝ ✟ ✠ ① On Hausdorff measure and differentiability Applications to submanifolds

On Hausdorff measure and We say that x ∈ Σ is algebraically regular if its differentiability Valentino homogeneous tangent space Ax Σ is a homogeneous Magnani subgroup of G. Area formulae for Lipschitz mappings

In fact, the deﬁnition of homogeneous tangent space is The Fractal purely algebraic and it can be given a priori on the point, Geometry inside homogeneous without knowing whether the blow-up limit exists. groups Measure theoretic area formula The existence of the blow-up limit (3) requires certain Applications to regularity conditions on the “manifold around x”. submanifolds

Conjecture If Σ is C1 smooth and x ∈ Σ is algebraically regular, then −1 + the blow-up limit δ1/r (x Σ) exists as r → 0 . On Hausdorff measure and differentiability Applications to submanifolds

On Hausdorff measure and differentiability

Valentino This conjecture can be proved to hold in all two step Magnani homogeneous groups [V. M., 2017, in progress]. Area formulae for Lipschitz In principle, algebraically regular points can be classified: mappings The Fractal each of them corresponds to a certain homogeneous Geometry inside homogeneous subgroup of G, defining a “specific Geometry”. groups Measure theoretic area We finally focus our attention on the simplest case, formula Applications to corresponding to a specific class of submanifolds, that submanifolds are everywhere tangent to a special smooth distribution of planes given by G. On Hausdorff measure and differentiability Applications to submanifolds

On Hausdorff measure and ι differentiability The grading of G = H1 ⊕ · · · H determines also a grading Valentino on its Lie algebra Lie(G) = V1 ⊕ · · · ⊕ Vι. Magnani A horizontal left invariant vector ﬁeld has the form Area formulae for Lipschitz mappings X(p) = dlpv for any p ∈ G The Fractal Geometry inside where v ∈ H1 ⊂ and l : → , x → px is the left homogeneous G p G G groups

translation. Then we define the first layer of Lie(G): Measure theoretic area formula V = {X ∈ Lie( ): X is horizontal} 1 G Applications to submanifolds We can now define the horizontal fiber

HpG = span {X(p): X ∈ V1} ⊂ TpG

for each p ∈ G. On Hausdorff measure and differentiability Applications to submanifolds

On Hausdorff measure and Horizontal submanifold differentiability 1 Valentino Σ is a C smooth horizontal submanifold if for each x ∈ Σ Magnani we have T Σ ⊂ H . x x G Area formulae for Lipschitz mappings

Examples The Fractal Geometry inside Horizontal curves are examples of one dimensional homogeneous horizontal submanifolds. Legendrian submanifolds in groups n Measure Heisenberg groups H are examples of higher theoretic area formula dimensional horizontal submanifolds. Applications to submanifolds Horizontal subgroup ι A horizontal subgroup H ⊂ G = H1 ⊕ · · · ⊕ H is a homogeneous subgroup such that H ⊂ H1. In particular it is a commutative subgroup. On Hausdorff measure and differentiability Applications to submanifolds

On Hausdorff measure and [From V. M., 2017, in progress] differentiability Valentino 1 If x ∈ Σ is algebraically regular and its homogeneous Magnani tangent space is a horizontal subgroup, then the blow-up Area formulae for limit exists. Lipschitz mappings 2 If Σ is a horizontal submanifold, then each of its points is The Fractal algebraically regular and the homogeneous tangent space Geometry inside homogeneous is everywhere a horizontal subgroup. groups

3 Measure Finally there holds theoretic area formula Hn(Σ) = Sn(Σ) Applications to submanifolds for all horizontal submanifolds.

The second statement follows from the h-differentiability of the parametrization of Σ. It is rather unexpected the fact that we take only the algebraic consequences of this differentiability, being the blow-up carried out using only the C1 smoothness.