Quasispheres and Bi-Lipschitz Parameterization

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1 Quasispheres and Bi-Lipschitz Parameterization Matthew Badger Stony Brook University September 15, 2012 This research was partially supported by DMS The speaker is currently supported by an NSF postdoctoral fellowship, NSF Grant DMS

2 Two Measurements of Distortion f : Ω Ω a homeomorphism of domains in R n, f W 1,n (Ω) If Df (x) exists and has eigenvalues λ 1 (x) λ n (x), then ( λ n (x) n K f (x) = max λ 1 (x) λ n (x), λ ) 1(x) λ n (x) λ 1 (x) n The maximal dilatation (local distortion) K f (Ω) = ess sup K f (x). x Ω The (weak) quasisymmetry constant (global distortion) { } f (y) f (x) y x H f (Ω) = max : x, y, z Ω, f (z) f (x) z x 1 Quasispheres and Bi-Lipschitz Parameterization Matthew Badger Stony Brook University 1 / 12

3 Local Distortion versus Global Distortion f : Ω Ω a homeomorphism of domains in R n, f W 1,n (Ω) For all n 2 and all domains Ω R n, K f (Ω) H f (Ω) n 1 When n 2 and Ω = R n, H f (R n ) 1 Φ n (K f (R n ) 1), Φ n : [0, ) [0, ) Precise formula for Φ 2 (t) was determined by Agard Φ n (0) = 0 (n 3) was proved by Vuorinen When n 2, Ω = R n and K is near 1, Φ 2 (K 1) C 2 (K 1) (n = 2) ( ) 1 Φ n (K 1) C n (K 1) log (n 3) K 1 Estimate (n 3) by Seittenranta 1996 (cf. Prause 2007) It is not known if the logarithm term is necessary. Quasispheres and Bi-Lipschitz Parameterization Matthew Badger Stony Brook University 2 / 12

4 Quasispheres A map f : R n R n is K-quasiconformal if f is a homeomorphism, f W 1,n (R n ), K f (R n ) K. A quasisphere f (S n 1 ) is image of S n 1 under global QC map. (A quasicircle f (S 1 ) is usual name for a quasisphere in the plane.) Examples Question: What is the relationship between the dilatation K f of f and the geometry of the quasisphere f (S n 1 )? Quasispheres and Bi-Lipschitz Parameterization Matthew Badger Stony Brook University 3 / 12

5 Dimension of Quasispheres A quasisphere f (S n 1 ) in R n has Hausdorff dimension d f = dim f (S n 1 ) where n 1 d f < n. If K f (R n ) = 1, then f is Möbius. Hence d f = n 1. Exist f (S n 1 ) with d f arbitrarily close to n. (Bishop 1999) d f n 1 + c n (K f (R n ) 1) (Mattila and Vuorinen 1990) d f n 1 + c n (K f (R n ) 1) (log 2 1 K f (R n ) 1 (Prause 2007): Idea! Exploit quasisymmetry H f of f. ) 2 Astala s conjecture : In the plane (n = 2), d f 1 + kf 2 k f = (K f (R 2 ) 1)/(K f (R 2 ) + 1). (Smirnov 2010) where Refined to H 1+k2 f (f (S n 1 )) < (Prause, Tolsa, Uriarte-Teuro 2012) Quasispheres and Bi-Lipschitz Parameterization Matthew Badger Stony Brook University 4 / 12

6 Asymptotically Conformal Quasispheres Natural to ask: To what degree is the geometry of f (S n 1 ) determined by the dilatation of f in a neighborhood S n 1? Let A t be annular neighborhood of S n 1 of size t. f (S n 1 ) is asymptotically conformal if K f (A t ) 1 as t 0. If K f (R n ) = 1, then d f = n 1 and H n 1 (f (S n 1 )) < If f (S n 1 ) is asymptotically conformal, then d f = n 1 There exist asymptotical conformal f (S n 1 ) such that H n 1 (f (S n 1 )) =. (e.g. flat snowflakes ) The problem is K f (A t ) can converge to 1 very slowly! Quasispheres and Bi-Lipschitz Parameterization Matthew Badger Stony Brook University 5 / 12

7 Rectifiability Let f : R n R n be quasiconformal and assume that t0 0 Ψ(K f (A t ) 1) dt t <. Theorem (Anderson, Becker, Lesley 1988) If Ψ(t) = t 2, n = 2 and f B(0,1) is conformal, then H 1 (f (S 1 )) <. Theorem (Mattila and Vuorinen 1990) If Ψ(t) = t, then f S n 1 is Lipschitz. Hence H n 1 (f (S n 1 )) <. Theorem (Reshetnyak 1994) If Ψ(t) = t, then f S n 1 is a C 1 embedded submanifold of R n. Quasispheres and Bi-Lipschitz Parameterization Matthew Badger Stony Brook University 6 / 12

8 Rectifiability (New Result!) Let f : R n R n be quasiconformal and assume that t0 0 Ψ(K f (A t ) 1) dt t <. Theorem (B., Gill, Rohde, Toro 2012) If Ψ(t) = ( t log t 1) 2, then H n 1 (f (S n 1 )) <. Moreover: f (S n 1 ) admits local (1 + δ)-bi-lipschitz parameterizations of R n 1 for every choice of δ > 0. This hypothesis includes the case Ψ(t) = t 2 ε, for any ε > 0 The exponent 2 is the best possible, cannot replace with 2 + ε. The conclusion is about f (S n 1 ), not about f S n 1. The conclusion is weaker than saying f (S n 1 ) is locally C 1 Quasispheres and Bi-Lipschitz Parameterization Matthew Badger Stony Brook University 7 / 12

9 True Version of Theorem: Quasisymmetry Version Let f : R n R n be quasiconformal and assume that t0 0 sup Υ(H f (B(x, t)) 1) dt x S n 1 t <. Theorem (B., Gill, Rohde, Toro 2012) If Υ(t) = t 2, then H n 1 (f (S n 1 )) <. Moreover: f (S n 1 ) admits local (1 + δ)-bi-lipschitz parameterizations of R n 1 for every choice of δ > 0. Proof is an observation of (Prause 2007) plus a theorem on existence of bi-lipschitz parameterizations (Toro ( 1995). ) Recall that H f (R n ) 1 C n (K f (R n ) 1) log. 1 K f (R n ) 1 To derive dilatation version of the theorem from quasisymmetry version, need to localize this estimate. Quasispheres and Bi-Lipschitz Parameterization Matthew Badger Stony Brook University 8 / 12

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11 Local Flatness of a Set Jones β-number: β A (x, r) = 1 r Local Flatness: θ A (x, r) = 1 r inf sup L G(n,n 1) y A B(x,r) dist(y, (x + L) B(x, r)) inf HD[A B(x, r), (x + L) B(x, r)] L G(n,n 1) 0 β A (x, r) θ A (x, r) 1 information when β or θ is small Quasispheres and Bi-Lipschitz Parameterization Matthew Badger Stony Brook University 10 / 12

12 Quasisymmetry Controls Local Flatness Lemma (Prause 2007, BGRT) If g : R n R n, g(±e 1 ) = ±e 1, then θ g(e 1 ) (0, 1 2 ) 20[H f (B 2 ) 1] How does this transfer to estimates on a quasisphere? θ f (Tx )(f (x), 1 4 f (e+ r ) f (e r )) 20[H f (B(x, 2r)) 1] where x S n 1 (red point), T x tangent plane, e ± r = x ± r n x (blue points) Quasispheres and Bi-Lipschitz Parameterization Matthew Badger Stony Brook University 11 / 12

13 Future Directions 1 In joint project with Jonas Azzam and Tatiana Toro, we are looking for conditions on H f which guarantee that f (S n 1 ) is uniformly rectifiable. 2 Converse? If f (S n 1 ) is a rectifiable quasisphere, must f (S n 1 ) = g(s n 1 ) for some quasiconformal map g : R n R n such that H g satisfies a square Dini condition? 3 Determine if it is possible to remove the logarithm term from the estimate H f (R n ) 1 C n (K f (R n 1 ) 1) log K f (R n ) 1. This is called the linear dilatation problem. Quasispheres and Bi-Lipschitz Parameterization Matthew Badger Stony Brook University 12 / 12