A floating body approach to Feffermans surface area David Barrett - - PDF document

A “floating body” approach to Fefferman’s surface area David Barrett Chapel Hill 25 October 2003

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Plan of talk (1) Real affine geometry (2) Complex analogue (3) My contribution (4) Questions / further directions

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Plane curves γ : I → R2 smooth ds = |γ′(t)| dt κ = ωarea (γ′(t),γ′′(t))

|γ′(t)|3

• γ

κp ds = . . .

• γ

3

√κ ds =

• I

3

• ω(γ′(t), γ′′(t)) dt def

= Aff(γ) = affine arc length of γ T : R2 → R2 affine ⇒ Aff(T (γ)) = det1/3(T ) · Aff(γ)

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Convex hypersurfaces in Rn S smooth ⊂ Rn Aff(S) =

• S

κ

1 n+1 dS = aff. surf. area of S

T : Rn → Rn affine ⇒ Aff(T (S)) = det

n−1 n+1(T ) · Aff(S)

What if S convex but not smooth? Use “convex floating bodies” – K ⊂ Rn convex body; δ > 0 Kδ

def

=

• H open half-space

vol(K∩H)≤δ

K ∩ H K \ Kδ = convex floating body

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Theorem [Blaschke / Leichtweiss / Sch¨ utt-Werner] If bK is C2 then Aff(bK) = lim

δց0 cn

vol(Kδ) δ2/(n+1). . . . but how does this help with non-smooth case? Definition/Theorem [Sch¨ utt-Werner] For K a general convex body, Aff(bK) = lim

δց0 cn

vol(Kδ) δ2/(n+1) =

• bK

κ

1 n+1

non-sing. dS.

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What does Aff(bK) tell us?

• Theorem [Gruber]. Let

FK(ν) = inf

P polyhedron⊂K # faces ≤ν

vol(K \ P ). Then FK(ν) ∼

ν→∞ cn Aff(bK)

n+1 n−1 · ν− 2 n−1.

Theorem [Blaschke / Hug]. Aff(bK)

n+1 n−1 · vol(K)−1 maximal for

ellipsoids.

• Corollary. Round potatoes are

hardest to peel.

• Theorem [B´

ar´ any / Sch¨ utt]. Prob (xν+1 / ∈ Hull{x1, . . . , xν}) ∼

ν→∞

cn Aff(bK) · vol(K)

1−n n+1 · ν− 2 n+1.

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Complex analysis Is there a complex analogue of affine surface area? S ⊂ Cn, smooth, (str.) pseudoconvex Definition [Fefferman, Adv. Math. ’79]. Fef(S) = cn

• S

| det L|

1 n+1 dS

T biholomorphic ⇒ integrand for Fef(T (S)) = det

2n n+1(T ′) · integrand for Fef(S)

⇒ corresponding Szeg¨

• kernels satisfy

SΩ(z, ζ) = SΦ(Ω)(Φ(z), Φ(ζ)) · det

n n+1Φ′(z) · det n n+1Φ′(ζ)

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What about non-smooth pseudconvex domains? → alt. def’s in the smooth case? Theorem [B.]. Let Ω ⊂⊂ Cn be str. ψ-convex, bΩ C3. For M > 0 let PM(Ω) be the set of holo. fcns. h on Ω s.t. (1) hC3(Ω) ≤ M; (2) ∅ = Ω ∩ h−1(0) ⊂ bΩ; (3) |dh| ≥ M−1 on Ω ∩ h−1(0). Let ΩM,δ =

• h∈PM(Ω), η>0

vol{|h|<η}<δ

{|h| < η}. Then for M large we have Fef(bΩ) = cn lim

δ→0

vol(ΩM,δ) δ1/(n+1) .

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Quasi-invariance of PM(Ω): if G : Ω1 → Ω2 a C3 holo. diffeo. ⇒ for M > 0 there are M♯ > M♭ > 0 s.t. PM♭(Ω2) ◦ G ⊂ PM(Ω1) ⊂ PM♯(Ω2) ◦ G. Why |h| rather than Re h?

• n = 1:

ΩM,δ ≈

• 2δ

π –collar of bΩ, so vol(ΩM,δ) √ δ

→

• 2

π ℓ(bΩ) = c1 Fef(bΩ)

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• n > 1:

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Sketch of proof (n=2):

• After vol.-pres. change of coords to

(z, w) = (z, u + iv), have h(z, w) =w Ω ={v > λ|z|2 + Re µz2 + . . . }

• vol(Ω∩{|w| < η}) = 2π

3 η3

√

λ2−|µ|2+O(η

7 2)

• The case µ = 0 occurs
• ΩM,δ has thickness ≈ C

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• δ|L|

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Questions / further directions (1) More general domains (2) Approx. by analytic polyhedra (3) Random holomorphic hulls (4) “Hausdorff-Levi” dimension Example: Ω = ∆ × ∆. Then limδ→0

vol(ΩM,δ)

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√ δ

= 0 but 0 < limδ→0

vol(ΩM,δ) √ δ

< ∞ leads to “correct” Hardy space. (5) Isoperimetric inequality: Is Fef bΩ · vol(Ω)−n/(n+1) maximized by affine balls?

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