The energy spaces of the tangent point energies

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1 The enegy spaces of the tangent point enegies Simon Blatt August 3, 2 In this small note, we will give a necessay and sucient condition unde which the tangent point enegies intoduced by Heiko von de Mosel and Pawel Stzelecki in [SvdM, SvdM] ae bounded. We show that an admissible submanifold has bounded E q -enegy if and only if it is injective and locally agees with the gaph of functions that belongs to Sobolev- Slobodeckij space W 2 m q,q. The known Moey embedding theoems of Heiko von de Mosel and Pawel Stzelecki can then be intepeted as standad Moey embedding theoem fo these spaces. Especially, this show that the Hölde exponents fo the embeddings in [SvdM] ae shap. Intoduction Vey ecently Heiko von de Mosel and Pawel Stzelecki stated the investigation of so called tangent point enegies fo cuves and sufaces and showed that they posses egulaizing and self epulsive eects fo a boad class of sets of abitay dimension and codimension which they called admissible sets. One of the main esults in this wok was that boundedness of these enegies fo an admissible set implies that this set function. Fo cuves they could give an example that shows that this Hölde exponent is optimal. Fo object of highe dimensions this was not known. The tangent point enegy of an m-ectiable set Σ R n is given by the double integal is locally a gaph of a C, q 2m E q Σ := Σ Σ R tp x, y q dh m xdh m y. Depatement Mathematik, ETH Züich, Rämistasse, CH-84 Züich, Switzeland, simon.blatt@math.ethz.ch

2 whee H m denotes the m-dimensional Hausdo measue and R tp x, y := x y 2 distx y, T x Σ fo all x y Σ such that the tangent space T x Σ on Σ exists. In this small note, we want to classify all admissible sets with nite tangent point enegy unde the assumption that the exponent q is bigge than the citical exponent 2m. As in the case of O'Haa's knot enegies cf. [Bla], it tuns out that this classication can be given with the help of Sobolev Slobodeckij spaces. Fo an open subset Ω R n and s,, p [, these ae dened using the seminoms f s,p := Ω Ω p fx fy p dxdy. x y n+sp Fo k N, the Sobolev-Slobodeckij space W k+s,p R m, R is the space of all functions f W k,p R m, R with α f s,p < fo all multiindices α N m with length α = k. The W k+,p -nom is then given by f W k,p := f W k,p + α N n α =k α f s,p. As usual, the space W k+s,p R m, R n is dened component wise. Moe infomation on these spaces can be found fo example in [Ada75, Ta7] The main esult of this aticle is the following Theoem.. Let Σ be a compact m-dimensional embedded C submanifold and q > 2m. Then E q Γ < if and only if Σ is an embedded W 2 m q,q submanifold. Hee an embedded W s,p submanifold fo s p m > is a submanifold which locally agees with the gaph of a W s,p function. Combining this with Theoem.4 in [SvdM] we get the following coollay Coollay.2. Let Σ be an admissible set and q > 2m. Then E q Σ < if and only if Σ is an embedded W 2 m q,q manifold. Especially, togethe with the shapness of the Moey embedding theoem this poves that the exponent m 2q in Theoem.4 of [SvdM] is optimal. In Section 2 we st give a poof of Theoem. fo cuves to illustate the main idea fo this simple toy poblem. Aftewads, with the help of some futhe techniques we pove the full statement in Section 3. 2

3 2 Poof fo cuves Of couse it is enough to show the theoem fo cuves of length. Let Γ C R/Z, R n, be a egula embedded cuve paametized by ac length and Σ = ΓR/Z. Then E q Γ := E q ΓR/Z = /2 R tp Γx, Γx + w q dxdw We chose a δ > such that R/Z /2 Γ x Γ y 2 2. fo all x, y R/Z with x y δ. If we denote by Π Γx the othogonal pojection of R n onto the nomal space of ΓR/Z at the point Γx, i.e. Π Γx v := v v, Γ x Γ x we get using Equation 2. that thee is a C < such that C Γ x Γ y Π Γx Π Γx Γ x Γ y fo all x, y R/Z with x y δ. Fom distγx Γy, T x Γ = Π Γx Γx Γy, one then sees that E q Γ = /2 R/Z /2 = 2 C C /2 R/Z /2 R/Z δ Π Γx Γx Γx + w q Γx Γx + w 2q dwdx δ δ R/Z δ Π Γx Γx Γx + w q + Π Γx+w Γx + w Γx q Γx Γx + w 2q dwdx Π Γx Π Γx+w q Γx Γx + w q dwdx Γ x Γ x + w q w q dwdx. Hence, /2 R/Z /2 Γ x + w Γ x q w q dxdw E q R/Z + δ q. 3

4 This implies that Γ W 2 q,q R/Z if E q Γ is nite. To get the othe implication, st note that evey embedded cuve Γ C R/Z, R n paametized by ac length satises a bi-lipschitz estimate, i.e. thee is a constant C = CΓ < such that w Γx + w Γx fo all w 2. Hence, E q Σ = /2 R/Z /2 /2 R/Z /2 Jensen's inequality ΠΓx Γx + w Γx q C Γx + w Γx q ΠΓx /2 R/Z /2 Γ x + τwdτ w dwdx q dwdx Γ s Γ s + τw q w q dwdsdτ As this implies Π Γx Γ y = Π Γx Γ y Γ x Γ y Γ x E q Σ R/Z 2 /2 Γ s Γ s + τw q w q Substituting w = τw, the ight hand side can be witten as τ p τ/2 R/Z τ/2 Γ s Γ s + w p w p Thus E q Σ is nite if Γ W 2 /q,q. = p d wdsdτ /2 R/Z /2 dwdsdτ Γ s Γ s + w p w p d wds <. 3 Poof fo submanifolds In the case of submanifolds, supisingly a simila idea as in the case of cuves woks though the technical details ae moe involved. 4

5 Fo m vectos v,..., v m R n, let A {vi} be the n m matix whose columns consist of v i and Π {vi} be the othogonal pojection onto the vecto space spanned by v i, i =,..., m. If the v,..., v m ae linealy independent, the nomal equations cf. [Sto99, pp ] lead to the epesentation and thus the map Π {vi} = A {vi} A {vi}a {v i} A {vi} v,..., v n Π {vi} is locally Lipschitz with espect to the Euclidean nom on R n m on the set {v i,..., v m R n m : v i ae linealy independent}. Now, let f C R m, R n m be such that f L, gx := fx, x, and let e,..., e m be a basis of R m. Then thee is a C such that fo any othogonal pojection P onto a m-dimensional subspace and any x R we have P P gx+ei gx sup i=,...,m Indeed, fo ε := ε e,..., e m > small enough, P id R mgx + e i gx. 3. P gx + e i gx gx + e i gx ε implies that the vectos P gx+e i gx ae linealy independent as thee pojections onto the st m coodinates ae linealy independent. Futhemoe, P gx + e i gx ε + 2 sup e i i=,...,m due to the Lipschitz estimate on f. Hence the local Lipschitz continuity poven above implies that if P Π {gx+ei gx} = Π {P gx+ei gx} Π {gx+ei gx} sup i=,...,m sup P id R mgx + e i gx ε. i=,...,m Since we always have P Π gx+ei gx 2, we deduce P id R mgx + e i gx P Π {gx+ei gx} C + 2/ε sup P id R ngx + e i gx i=,...,m 5

6 which poves Equation 3. To pove the theoem, let ẽ i, i =,..., m be an othonomal basis of R m and let { ẽ, if j = e j := 2 ẽ + ẽ j if j = 2,..., m. Since e i, i =,..., m is a basis of R m, we get and scaling leads to P Π gx+ei gx sup i=,...,m P id R mgx + e i gx. P Π gx+e i gx sup gx + P id R m ei gx i=,...,m gx + P id ẽ gx + gx + P id 2 ẽ + ẽ i gx fo all. Since both sides of this inequality ae invaiant unde othogonal tansfomation of the ẽ i, the constant in it does not depend on the choice of the othonomal basis ẽ,..., ẽ m. Exchanging ẽ with ẽ and x with x + ẽ and obseving that this leaves the vecto space spanned by gx + e i gx invaiant, we futhemoe get that fo all othogonal pojections Q of R n onto an m-dimensional subspace we have Q Π gx+e i gx gx gx + Q id ẽ + gx + Q id 2 ẽ + ẽ i gx + ẽ and hence P Q gx + P id ẽ gx + gx + P id 2 ẽ + ẽ i gx +C gx gx + Q id ẽ + gx + Q id 2 ẽ + ẽ i gx + ẽ fo all othogonal pojections P, Q of R n onto m-dimensional subspaces. Now let Σ be an m-dimensional C submanifold with nite E q enegy and x Σ. Afte some otation and tanslation we can assume that x = and T x Σ = R m {} and that thee is an > and an f C R m, R n m with f = such that 3.2 f L 3.3 6

7 and gb Σ. Let Π gx denote the othogonal pojection of R n onto the tangent space on gr m at gx. Fo x B /2, /2 and any othonomal basis e,... e m of R m we have by the above discussion Π gx + Π gx+e gx gx + Π ẽ gx id + C gx + Π 2 gx id ẽ + ẽ i gx + ẽ + C gx + Π ẽ gx gx+e id + C gx + Π 2 gx+e id ẽ + ẽ i gx R tp gx, gx + e gx gx + ẽ 2 + C R tp gx + 2 e + e i, gx + e gx + 2 ẽ + ẽ i gx + ẽ 2 + CR tp gx + e, gx gx + e gx 2 + C R tp gx + 2 e + e i, gx + e gx + 2 ẽ + ẽ i gx + ẽ 2 Using that due to the Lipschitz bound on f we have that gx gy 2 x y, we deduce Π gx Π gx+e q q R tp gx, gx + e q + C R tp gx + 2 e + e i, gx + e q + CR tp gx + e, gx q + C R tp gx + 2 e + e i, gx + e q. To get id of the dependence on a special othonomal basis, we conside the space of othonomal matices Om as a m := mm 2 dimensional submanifold of R m m and the fact that fo a function f L S m, R we have fa i dh m A = H m Om H m S m fωdh m ω. A Om S m 7

8 This leads to B /2;R m B /2;R m Π x Π x+w q w q dh m wdh m x = B /2;R m = Cm /2 B /2;R m B /2;R m B /2;R m E q Σ. Togethe with the estimate + m /2 /2 S m m m Π x Π x+e q w q dh m eddh m x Om Om Π x Π x+a q w q R tp gx, gx + A q R tp g x + 2 A + A i, gx + A dh m AdH m w q + R tp gx + A, gx q + R tp g x + q 2 A + A i, gx + A dh m AddH m x /2 S m R tp gx, gx + e q + m R tp g x + 2 /2 e, gx q q + R tp gx + e, gx + m R tp g x + 2 /2 e q, gx dh m eddh m x f x f y Π gx Π gy this poves hat Σ is a W 2 q,q submanifold if E q Σ is nite. Do get the othe implication, we use that if Σ is a compact W 2 /q,q manifold thee is an > such that fo all x in Σ thee is a function f W 2 /q,q R m, R n m with f = such that f L 8

9 and an A SOn such that the function gx := fx, x satises Fo xed x Σ we then use and to get B /2 B /2 AgB /2; R m + x Σ. Π gx Π gy f x f y, gx gy x y Π gx gx + w gx q gx + w gx 2q dh m wdh m x B /2 B /2 B /2 B /2 B /2 B /2 Π gx g x + τw q w q dτdh m wdh m x Π gx g x + τw g x q w q dτdh m wdh m x g x + τw g x q w q dτdh m wdh m x. Substituting w = τw this expession can be estimated as in the case of cuves by g x + w g x q C w q dh m wdh m x f 2 W q,q B B /2 B /2 Let d Σ denote the geodesic distance Σ and B x; Σ denote the geodesic ball of adius in Σ. Then the above calculation combined with the fact that the gadient of f is bounded leads to B /2x,Σ y Σ d Σ x,y 2 fo all x Σ. R tp x, y q dhm yh m x B /2 B /2 Π gx gx + w gx q gx + w gx 2q dh m wdh m x < 9

10 Since Σ is compact an easy coveing agument then gives R tp x, y q dhyhm x < y,x Σ d Σ x,y /4 Since Γ is an embedded C manifold we futhemoe have that { } x y µ := inf d Σ x, y : x, y Σ, d Σx, y > /4 > and hence we nally get E q Σ = Σ x Σ,d Σx,y /4 + R tp x, y q dhm yh m x Σ x Σ,d Σx,y /4 Σ x Σ,d Σx,y /4 This nishes the poof of the theoem. Refeences [Ada75] R tp x, y q dhm yh m x R tp x, y q dhm yh m x + µ q Hm Σ <. Robet A. Adams. Sobolev spaces. Academic Pess [A subsidiay of Hacout Bace Jovanovich, Publishes], New Yok-London, 975. Pue and Applied Mathematics, Vol. 65. [Bla] Simon Blatt. Boundedness and egulaizing eects of O'Haa's knot enegies. Accepted fo publication by the Jounal of Knot Theoy and its Ramication, 2. [Sto99] Josef Stoe. Numeische Mathematik. : Eine Einfühung - unte Beücksichtigung von Volesungen von F. L. Baue. 8., neu beab. u. ew. Au. Belin: Spinge, 999. [SvdM] Pawel Stzelecki and Heiko von de Mosel. enegies fo cuves. June 2. Tangent-point self-avoidance [SvdM] Pawel Stzelecki and Heiko von de Mosel. Tangent-point epulsive potentials fo a class of non-smooth m-dimensional sets in n. pat i: Smoothing and self-avoidance eects. Febuay 2. [Ta7] Luc Tata. An intoduction to Sobolev spaces and intepolation spaces, volume 3 of Lectue Notes of the Unione Matematica Italiana. Spinge, Belin, 27.

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