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 1

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

Plane curves γ : I → R 2 smooth ds = | γ ′ ( t ) | dt κ = ω area ( γ ′ ( t ) ,γ ′′ ( t )) | γ ′ ( t ) | 3 κ p ds = . . . � γ √ κ ds = ω ( γ ′ ( t ) , γ ′′ ( t )) dt def 3 � 3 � � = Aff( γ ) γ I = affine arc length of γ T : R 2 → R 2 affine ⇒ Aff( T ( γ )) = det 1 / 3 ( T ) · Aff( γ ) 3

Convex hypersurfaces in R n S smooth ⊂ R n 1 n +1 dS = aff. surf. area of S � Aff( S ) = κ S T : R n → R n affine n − 1 ⇒ Aff( T ( S )) = det n +1 ( T ) · Aff( S ) What if S convex but not smooth? Use “convex floating bodies” – K ⊂ R n convex body; δ > 0 def � K δ = K ∩ H H open half-space vol( K ∩ H ) ≤ δ K \ K δ = convex floating body 4

Theorem [ Blaschke / Leichtweiss / utt-Werner ] If b K is C 2 then Sch¨ vol( K δ ) Aff(b K ) = lim δ ց 0 c n δ 2 / ( n +1) . . . . but how does this help with non-smooth case? Definition/Theorem [ Sch¨ utt-Werner ] For K a general convex body, vol( K δ ) Aff(b K ) = lim δ ց 0 c n δ 2 / ( n +1) 1 � n +1 = κ non-sing. dS. b K 5

What does Aff(b K ) tell us? • Theorem [ Gruber ]. Let F K ( ν ) = inf vol( K \ P ) . P polyhedron ⊂ K # faces ≤ ν Then n +1 2 n − 1 · ν − F K ( ν ) ν →∞ c n Aff( b K ) ∼ n − 1 . Theorem [ Blaschke / Hug ]. n +1 n − 1 · vol( K ) − 1 maximal for Aff( b K ) ellipsoids. Corollary. Round potatoes are hardest to peel. • Theorem [ B´ ar´ any / Sch¨ utt ]. Prob ( x ν +1 / ∈ Hull { x 1 , . . . , x ν } ) ∼ ν →∞ 1 − n 2 n +1 · ν − c n Aff(b K ) · vol( K ) n +1 . 6

Complex analysis Is there a complex analogue of affine surface area? S ⊂ C n , smooth, (str.) pseudoconvex Definition [ Fefferman, Adv. Math. ’79 ]. � 1 n +1 dS Fef( S ) = c n | det L | S T biholomorphic ⇒ integrand for Fef( T ( S )) 2 n n +1 ( T ′ ) · integrand for Fef( S ) = det ⇒ corresponding Szeg¨ o kernels satisfy S Ω ( z, ζ ) = S Φ(Ω) (Φ( z ) , Φ( ζ )) n n n +1 Φ ′ ( z ) · det n +1 Φ ′ ( ζ ) · det 7

What about non-smooth pseudconvex domains? → alt. def’s in the smooth case? Theorem [ B. ]. Let Ω ⊂⊂ C n be str. ψ -convex, b Ω C 3 . For M > 0 let P M (Ω) be the set of holo. fcns. h on Ω s.t. (1) � h � C 3 (Ω) ≤ M ; (2) ∅ � = Ω ∩ h − 1 (0) ⊂ b Ω ; (3) | dh | ≥ M − 1 on Ω ∩ h − 1 (0) . � {| h | < η } . Let Ω M,δ = h ∈ P M (Ω) , η> 0 vol {| h | <η } <δ Then for M large we have vol(Ω M,δ ) Fef( b Ω) = c n lim δ 1 / ( n +1) . δ → 0 8

Quasi-invariance of P M (Ω): if G : Ω 1 → Ω 2 a C 3 holo. diffeo. ⇒ for M > 0 there are M ♯ > M ♭ > 0 s.t. P M ♭ (Ω 2 ) ◦ G ⊂ P M (Ω 1 ) ⊂ P M ♯ (Ω 2 ) ◦ G . Why | h | rather than Re h ? • n = 1: � 2 δ Ω M,δ ≈ π –collar of bΩ, so vol(Ω M,δ ) � 2 → √ π ℓ (bΩ) = c 1 Fef(bΩ) δ 9

• n > 1: 10

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 µz 2 + . . . } η 3 7 • vol(Ω ∩{| w | < η } ) = 2 π √ λ 2 −| µ | 2 + O ( η 2 ) 3 • The case µ = 0 occurs � 3 • Ω M,δ has thickness ≈ C δ | L | 11

Questions / further directions (1) More general domains (2) Approx. by analytic polyhedra (3) Random holomorphic hulls (4) “Hausdorff-Levi” dimension Example: Ω = ∆ × ∆. Then vol(Ω M,δ ) lim δ → 0 = 0 but √ 3 δ vol(Ω M,δ ) < ∞ leads 0 < lim δ → 0 √ δ to “correct” Hardy space. (5) Isoperimetric inequality: Is Fef bΩ · vol(Ω) − n/ ( n +1) maximized by affine balls? 12