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Support Functions, Projections and Minkowski Addition of Legendrian Cycles

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Support Functions, Projections and Minkowski Addition of Legendrian Cycles SUPPORT FUNCTIONS, PROJECTIONS AND MINKOWSKI ADDITION OF LEGENDRIAN CYCLES ANDREAS BERNIG n n−1 Abstract. The structure of the space LC(R × S ) of compactly n n−1 supported integral Legendrian cycles on R × S is studied using Geometric Measure Theory. To each such cycle T , a support function 1 n−1 hT ∈ L (S , Z[R]) is constructed. For almost all k-dimensional linear subspaces L, a Legendrian cycle πL(T ), whose support function is the restriction of the support function of T to L ∩ Sn−1, is constructed. n n−1 It is shown that there is a partial ring structure on LC(R × S ), the multiplication being a generalized Minkowski addition. An example of non-existence of the Minkowski sum is given, but it is shown that the Minkowski sum does exist after changing one of the summands by an arbitrarily small linear map. Finally, mean projection formulas are formulated and proved in the context of Legendrian cycles. Introduction Legendrian cycles appear naturally in the study of geometric and topo- n logical properties of singular spaces. Subsets of R belonging to various geometrically interesting classes, e.g. piecewise linear spaces, convex and polyconvex sets, submanifolds, sets with positive reach, subanalytic sets, o-minimal sets and certain Lipschitz submanifolds, can be studied by asso- n ciating Legendrian cycles in SR , called normal cycles. We refer the reader to the survey [1] for more information and references concerning these con- structions and for applications in Differential and Integral Geometry. n The aim of this paper is to understand the structure of the space LC(SR ) n of compactly supported rectifiable Legendrian cycles on SR . Using slicing n theory, we define the support function hT of a cycle T ∈ LC(SR ). This support function generalizes the classical support function for compact con- 1 vex sets and is an L -function with values in the group ring Z[R]. Fu’s uniqueness theorem can be used to show that the support function yields n 1 N−1 an embedding of LC(SR ) ,→ L (S , Z[R]). Our first main result is the existence of an almost everywhere defined projection operator πL. For the notation, see Section 1. The Euler charac- teristic χ(T ) of a Legendrian cycle T is defined in Proposition 1.4 n Theorem 1. Let T ∈ LC(SR ), 1 ≤ k ≤ n. Then for almost all L ∈ Gro(n, k) there exists a unique Legendrian cycle πL(T ) ∈ LC(SL) whose support function equals the restriction of the support function of T to S(L). MSC: 49Q15, 53C65, 14P10 Key words: Legendrian currents, subanalytic sets, support function, Minkowski addition, Lipschitz-Killing invariants Supported by DFG-grant BE2484/2-1 . 1 2 ANDREAS BERNIG The functionals T 7→ πL(T ) are Lipschitz (with respect to mass M and to flat norm F) in the average, i.e. there is a constant C = C(n, k) with Z M(πL(T ))dµn,k(L) ≤ C(n, k)M(T ) Gro(n,k) Z F(πL(T ))dµn,k(L) ≤ C(n, k)F(T ). Gro(n,k) For almost all L, χ(πL(T )) = χ(T ). The ring structure of Z[R] is used to define the Minkowski sum of two Legendrian cycles, which generalizes the classical Minkowski sum of compact convex sets. If the Minkowski sum of two Legendrian cycles exists (which is not always the case, as it is shown by an example), then its support function is the product of the support functions of the summands. n n The group GL(R ) acts in a natural way on LC(SR ). We show that after deforming one of the two summands by a linear map, the Minkowski sum exists: n n Theorem 2. Let T1,T2 ∈ LC(SR ). Then for almost all g ∈ GL(R ), the Minkowski sum T1 ⊕ g(T2) exists. To each Legendrian cycle T , one can associate real invariants Λ0(T ),..., Λn(T ), called Lipschitz-Killing invariants. We use our theory to obtain projection formulas for them. It seems that even in the subanalytic case, the corre- sponding formulas were not yet stated explicitly, although a direct proof would be easy using the linear kinematic formula. n Theorem 3. Let T ∈ LC(SR ) and 0 ≤ i ≤ k ≤ n. Then Z Λi(πL(T ))dµn,k = Λi(T ). (1) Gro(n,k) The notion of Minkowski sum (under the name Euler multiplication) was recently studied by Br¨ocker ([4]) in the subanalytic context. This case is simpler, since subanalyticity implies that the Euler multiplication is always defined. He also considered the projection map (push-forward), again it is always defined. Our approach, which is completely different and relies on Geometric Mea- sure Theory instead of Subanalytic Geometry, has the advantage of yielding results for all classes considered above, including sets with positive reach and polyconvex sets. Besides that, it uncovers some hidden structure on the space of Legendrian cycles, which is interesting in itself. Thanks. I would like to thank Ludwig Br¨ocker, Georges Comte and Daniel Hug for interesting discussions. 1. Definition of the support transform Notation. Besides standard notation from Geometric Measure Theory (as in [7]), we will use the following notation. The unit sphere in a Euclidean vector space V will be denoted by S(V ). The sphere bundle is given by SV := V × S(V ). LEGENDRIAN CYCLES 3 n The space RK (SR ) of i.m. rectifiable currents with support contained in a fixed compact set K admits a norm FK defined by FK (T ) := inf{M(A) + n M(B): T = A + ∂B, A, B ∈ RK (SR )}. We also define on the space n Rcpt(SR ) of compactly supported i.m. rectifiable currents F(T ) := inf{M(A)+ n M(B): T = A + ∂B, A, B ∈ Rcpt(SR )}. If K is the product of a compact n n−1 convex subset of R with S , then there exists a 1-Lipschitz retraction of n n SR to K and F(T ) = FK (T ) for T ∈ RK (SR ). n n−1 The sphere bundle R × S is a contact manifold with canonical 1- Pn n n−1 n n form α = i=1 yidxi (here we used the embedding R × S ,→ R × R n n and canonical coordinates (x1, . , xn, y1, . , yn) on R × R ). A current n T ∈ Dn−1(SR ) is called Legendrian if it annihilates α, i.e. T xα = 0. n Our object of study will be the space LC(SR ) of compactly supported n n n−1 integer-multiplicity rectifiable Legendrian cycles on SR = R × S . If n n K ⊂ SR is compact, we will write LCK (SR ) for the subspace consisting of currents with support in K. We remark that a similar notion appeared in [13] under the name general normal cycle. The group ring Z[R] can be naturally identified with the space I0(R) of integer multiplicity rectifiable 0-currents on R and inherits the flat norm. Note that R is naturally embedded in Z[R] and that the augmentation map in Z[R] corresponds to the evaluation at 1 in I0(R). The L1-norm and the F-norm of an almost everywhere defined function n−1 R h : S → Z[R] are defined by khkL1 := Sn−1 M(h(v))dv and khkF := R 1 Sn−1 F(h(v))dv respectively. The corresponding space of L -functions is 1 n−1 n−1 L (S , Z[R]) := {h : S → Z[R], khkL1 < ∞}/ ∼, where h1 ∼ h2 if and only if h1(v) = h2(v) for almost all v. n n−1 Definition 1.1. Let u : R × S → R, (x, v) 7→ hx, vi. Define the support n n−1 function of T ∈ LC(SR ) by hT (v) := u∗hT, π, vi ∈ I0(R) for all v ∈ S such that the slice is well-defined and i.m. rectifiable. Example 0 • Let hK (v) := supx∈K hx, vi denote the classical support function of a n compact convex set K ⊂ R . Then the support function of its normal cycle K˜ is given by h ˜ (v) = δ 0 ∈ I0( ). K hK (v) R n • Let M ⊂ R be a smooth, compact submanifold. For almost all n−1 n v ∈ S , the restriction of the height function R → R, x 7→ hx, vi on P M is a Morse function. Then hM˜ (v) = j njδtj where tj runs through the critical values of the height function and nj = ±1 according to whether the corresponding Morse index is even or odd. n • Br¨ocker defined a support function for subanalytic sets X ⊂ R ([4]). It follows readily from Fu’s uniqueness theorem that, in the compact case, it coincides a.e. with the support function of its normal cycle. We can describe the support function as in the case of submanifolds using (stratified) Morse theory ([12], [5]). In the subanalytic case, almost each height function is a stratified Morse function on X. Then P hX˜ (v) = j njδtj , where tj runs through the critical values of the height function and nj is the corresponding Morse index (which is a natural number that can be different from ±1). 4 ANDREAS BERNIG n Proposition 1.2. Let K ⊂ SR be compact. a) There exists a constant C(K) > 0 such that khT kL1 ≤ C(K)M(T ) n 1 n−1 for all T ∈ LCK (SR ). In particular, hT ∈ L (S , Z[R]). b) There exists a constant C(K) > 0 such that khT kF ≤ C(K)F(T ) for n all T ∈ LCK (SR ). n 1 n−1 c) The support transform LC(SR ) → L (S , Z[R]) is additive and injective.
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