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Isoperimetric Inequalities and the Structure of Metric Spaces - Part 2

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Isoperimetric Inequalities and the Structure of Metric Spaces - Part 2 Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Isoperimetric inequalities and the structure of metric spaces - Part 2 Stefan Wenger University of Fribourg July 20, 2013 Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Overview - Part 2 1 Review and aim of today’s lecture 2 Brief review of currents in metric spaces 3 Asymptotic cones of metric spaces 4 Persistence of quadratic isoperimetric inequality Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Review and aim X metric space, c : S1 ! X Lipschitz curve Fillarea0(c) = inf fArea('): ': D ! X Lip, 'jS1 = cg : Filling area function in X : X FA0 (r) = sup fFillarea0(c): L(c) ≤ rg. Aim is to prove Theorem (W.) X geodesic metric space. If there exists " > 0 such that 1 − " FAX (r) ≤ r 2 8r 1 0 4π X then FA0 (r) r. Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality . and . Theorem (W.) G α There exists nilpotent Lie group G of step 2 such that FA0 (r) 6' r for any α 2 R. Basic idea in both proofs: 1 Rescale metric d in X by factors rn & 0. Sequence (X ; rn d) has limit asymptotic cones X!. 2 Would like: X 2 X! 2 FA0 (r) r ) FA0 (r) ≤ Cr 8r ≥ 0. X! X! True? True for homological version FA (r) of FA0 (r). Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Ambrosio-Kirchheim currents: m-forms in metric spaces X complete metric space Idea (De Giorgi): Use (m + 1)-tuples (f ; π1; : : : ; πm) of Lipschitz functions on X as substitute for m-forms. N If X = R and f ; πi smooth then think (f ; π1; : : : ; πm) ! f dπ1 ^ · · · ^ dπm: Note: d(f dπ1 ^ · · · ^ dπm) = df ^ dπ1 ^ · · · ^ dπm If ' : RM ! RN smooth then ∗ ' (f dπ1 ^ · · · ^ dπm) = f ◦ ' d(π1 ◦ ') ^ · · · ^ d(πm ◦ ') Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Definition For X complete metric space and m ≥ 0: m m D (X ) := Lipb(X ) × Lip(X ) : Notation: Lip(X ) := ff : X ! R : f Lipschitzg Lipb(X ) := ff : X ! R : f Lipschitz and boundedg B1(X ) := ff : X ! R : f Borel and boundedg For f : X ! Y set 0 dY (f (x); f (x )) Lip(f ) := sup 0 x6=x0 dX (x; x ) Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Currents in metric spaces (Ambrosio-Kirchheim) Definition: A function T : Dm(X ) ! R is called m-current if: (i) T is multi-linear. n n n (ii) If πi 2 Lip(X ) with supn Lip(πi ) < 1 and πi ! πi then n n T (f ; π1 ; : : : ; πm) ! T (f ; π1; : : : ; πm): (iii) If πi is constant on spt f for some i then T (f ; π1; : : : ; πm) = 0: (iv) 9 finite Borel measure µ, concentrated on σ-cpt set, such that m Y Z jT (f ; π1; : : : ; πm)j ≤ Lip(πi ) jf jdµ i=1 X m for all (f ; π1; : : : ; πm) 2 D (X ). Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Space of m-currents in X : Mm(X ) := m-currents in X : Remark: Each T 2 Mm(X ) extends uniquely to T : B1(X ) × Lip(X )m ! R: Can show it satisfies same properties (i) – (iv) with f 2 B1(X ). Proposition Given T 2 Mm(X ) there exists a smallest Borel measure on X satisfying (iv), denoted kT k. For U ⊂ X open n X n n X o kT k(U) = sup jT (fn; π )j : Lip(πi ) ≤ 1; jfnj ≤ 1U : n Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Define mass of T by M(T ) := kT k(X ). Lower semi-continuity of mass: Corollary If (Tn) ⊂ Mm(X ) converges weakly (i.e. pointwise) to T 2 Mm(X ) then M(T ) ≤ lim inf M(Tn): n!1 Remark: Mm(X ) becomes a complete metric space with dM (T ; S) := M(T − S): Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Constructions for currents Constructions for T 2 Mm(X ): (i) If m ≥ 1 define @T (f ; π1; : : : ; πm−1) := T (1; f ; π1; : : : ; πm−1): ) @T satisfies axioms (i) – (iii) for currents; and @@T = 0. (ii) If ' : X ! Y Lipschitz define '#T (g; τ1; : : : ; τm) := T (g ◦ '; τ1 ◦ '; : : : ; τm ◦ '): m ) '#T 2 Mm(Y ) and k'#T k ≤ Lip(') '#kT k. (iii) For A ⊂ X Borel define (T A)(f ; π1; : : : ; πm) := T (f 1A; π1; : : : ; πm): ) T A 2 Mm(X ) and kT Ak = kT k A. Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Normal currents Example: For θ 2 L1(Rm) Z [θ](f ; π) := f θ det(dπ)dL m Rm defines metric m-current in Rm and k[θ]k = jθjdL m. Space of normal m-currents in X : Nm(X ) := T 2 Mm(X ): @T 2 Mm−1(X )g: Theorem If X is compact and (Tn) ⊂ Nm(X ) satisfies sup [M(Tn) + M(@Tn)] < 1 n then there exists (Tnj ) converging weakly to some T 2 Nm(X ). Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Integer rectifiable and integral currents Definition T 2 M0(X ) is called integer rectifiable if 9xi 2 X and mi 2 Z, i = 1;:::; n, with n X T = mi [xi ]: i=1 Definition T 2 Mm(X ) with m ≥ 1 is called integer rectifiable if (i) kT k is concentrated on a countably H m-rectifiable set. (ii) kT k vanishes on H m-negligible Borel sets. (iii) 8' : U ! Rm Lipschitz, U ⊂ X open, 9θ 2 L1(Rm; Z) with '#(T U) = [θ]: Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Space of integer rectifiable m-currents in X : Im(X ) := T 2 Mm(X ): T integer rectifiableg: Representation of integer rectifiable currents: Theorem m If T 2 Im(X ) with m ≥ 1 then 9 i : Ki ! X biLipschitz, Ki ⊂ R 1 cpt, and 9 θi 2 L (Ki ; Z) such that 1 1 X X T = i#[θi ] and M(T ) = M( i#[θi ]): i=1 i=1 Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Integral currents and Closure Theorem Space of integral m-currents in X : Im(X ) := Im(X ) \ Nm(X ): Closure Theorem: Theorem If T 2 Nm(X ) is the weak limit of a sequence (Tn) ⊂ Im(X ) with sup [M(Tn) + M(@Tn)] < 1 n then T 2 Im(X ). Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Ultralimits of metric spaces How to define limit of (arbitrary) sequence of metric spaces? Let (Xn) be sequence of metric spaces Xn = (Xn; dn). For given base points pn 2 Xn define n o X^ = (xn)n2N : xn 2 Xn, sup dn(xn; pn) < 1 : n 0 ^ For (xn); (xn) 2 X want to define distance by 0 lim dn(xn; x ) n!1 n if limit exists. 9 device making consistent "choice" of convergent subsequences: Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality Non-principal ultrafilters Let ! be a non-principal ultra-filter on N: ! : 2N ! f0; 1g finitely additive; !(N) = 1 and !(A) = 0 for all A ⊂ N finite. Existence: Zorn’s lemma. Proposition If Z is compact Hausdorff and (zn) ⊂ Z then 9! z 2 Z such that for all U ⊂ Z open with z 2 U !(fn 2 N : zn 2 Ug) = 1: Notation: lim! zn := z. Stefan Wenger Isoperimetric inequalities and metric structures Review and aim of today’s lecture Brief review of currents in metric spaces Asymptotic cones of metric spaces Persistence of quadratic isoperimetric inequality 0 ^ For (xn); (xn) 2 X define 0 0 d!((xn); (xn)) := lim! dn(xn; xn).
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