GMT – Varifolds Cheat-sheet Sławomir Kolasiński Some notation [id & cf] The identity map on X and the characteristic function of some E ⊆ X shall be denoted by idX and 1E : [Df & grad f] Let X, Y be Banach spaces and U ⊆ X be open. For the space of k times continuously k differentiable functions f ∶ U → Y we write C (U; Y ). The differential of f at x ∈ U is denoted Df(x) ∈ Hom(X; Y ) : In case Y = R and X is a Hilbert space, we also define the gradient of f at x ∈ U by ∗ grad f(x) = Df(x) 1 ∈ X: [Fed69, 2.10.9] Let f ∶ X → Y . For y ∈ Y we define the multiplicity −1 N(f; y) = cardinality(f {y}) : [Fed69, 4.2.8] Whenever X is a vectorspace and r ∈ R we define the homothety µr(x) = rx for x ∈ X: [Fed69, 2.7.16] Whenever X is a vectorspace and a ∈ X we define the translation τ a(x) = x + a for x ∈ X: [Fed69, 2.5.13,14] Let X be a locally compact Hausdorff space. The space of all continuous real valued functions on X with compact support equipped with the supremum norm is denoted K (X) : [Fed69, 4.1.1] Let X, Y be Banach spaces, dim X < ∞, and U ⊆ X be open. The space of all smooth (infinitely differentiable) functions f ∶ U → Y is denoted E (U; Y ) : The space of all smooth functions f ∶ U → Y with compact support is denoted D(U; Y ) : It is endowed with a locally convex topology as described in [Men16, Definition 2.13]. (Multi)linear algebra Let V; Z be vectorspaces. [Fed69, 1.4.1] The vectorspace of all k-linear anti-symmetric maps ' ∶ V × ⋯ × V → Z shall be denoted by k ⋀ (V; Z) : k k In case Z = R, we write ⋀ V = ⋀ (V; R). Last update: November 6, 2017 Page 1 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński k th [Fed69, 1.3.1] A vectorspace W together with µ ∈ ⋀ (V; W ) is the k exterior power of V if for any k vectorspace Z and ' ∈ ⋀ (V; Z) there exists a unique linear map '~ ∈ Hom(W; Z) such that ' = '~ ○ µ. µ V × ⋯ × V / W ∃!' ~ ∀' % Z We shall write W = ⋀kV and µ(v1; : : : ; vk) = v1 ∧ ⋯ ∧ vk : k We shall frequently identify ' ∈ ⋀ (V; Z) with '~ ∈ Hom(⋀kV; Z). [Fed69, 1.3.2] If V = span{v1; : : : ; vm}, then ⋀kV = spanvλ(1) ∧ ⋯ ∧ vλ(k) ∶ λ ∈ Λ(m; k) = spanvλ ∶ λ ∈ Λ(m; k) ; where Λ(m; k) = {λ ∶ {1; 2; : : : ; k} → {1; 2; : : : ; m} ∶ λ is increasing}. [Fed69, 1.3.1] If f ∈ Hom(V; Z), then ⋀kf ∈ Hom(⋀kV; ⋀kZ) is characterised by ⋀kf(v1 ∧ ⋯ ∧ vk) = f(v1) ∧ ⋯ ∧ f(vk) for v1; : : : ; vk ∈ V: [Fed69, 1.3.4] If f ∈ Hom(V; V ) and dim V = k < ∞, then ⋀kV ≃ R. We define the determinant det f ∈ R of f by requiring ⋀kf(v1 ∧ ⋯ ∧ vk) = (det f)v1 ∧ ⋯ ∧ vk ; whenever v1; : : : ; vk is a basis of V . [Fed69, 1.4.5] If f ∈ Hom(V; V ) and dim V = k < ∞ and v1; : : : ; vk is basis of V and !1;:::;!k is the 1 dual basis of ⋀ V = Hom(V; R), then we define the trace of f, denoted tr f, by setting k tr f = Q !i(f(vi)) ∈ R : i=1 [Fed69, 1.7.5] If V is equipped with a scalar product (denoted by ●) and {v1; : : : ; vm} is an orthonormal basis of V , then ⋀kV is also equipped with a scalar product such that {vλ ∶ λ ∈ Λ(m; k)} is orthonormal. In particular, tr(⋀kf) = Q ⋀kf(vλ) ● vλ : λ∈Λ(m;k) [Fed69, 1.7.2] Orthogonal injections are maps f ∶ X → Y between inner product spaces such that f(x) ● f(y) = x ● y whenever x; y ∈ X. We set m n m O(n; m) = {j ∈ Hom(R ; R ) ∶ ∀x; y ∈ R j(x) ● j(y) = x ● y} : [Fed69, 1.7.4] Orthogonal projections are maps f ∶ Y → X between finite dimensional inner product ∗ 1 1 spaces, such that f ∶ ⋀ X → ⋀ Y is an orthogonal injection. We set ∗ ∗ O (n; m) = {j ∶ j ∈ O(m; n)} : In case n = m we write ∗ O(n) = O (n; n) = O(n; n) : Last update: November 6, 2017 Page 2 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński [Fed69, 1.7.4] If V , Z are equipped with scalar products and f ∈ Hom(V; Z), then the adjoint map ∗ ∗ f ∈ Hom(Z; V ) is defined by the identity f(v) ● z = v ● f (z) for v ∈ V and z ∈ Z. We define the (Hilbert-Schmidt) scalar product and norm in Hom(V; Z) by setting for f; g ∈ Hom(V; Z) ∗ 1~2 f ● g = tr(f ○ g) and SfS = (f ● f) : [Fed69, 1.7.6] If f ∶ X → Y is an orthogonal injection [projection], then so is ⋀kf ∶ ⋀kX → ⋀kY . [Fed69, 1.7.6] If V , Z are equipped with norms, then the operator norm of f ∈ Hom(V; Z) is YfY = sup{Sf(v)S ∶ v ∈ V; SvS ≤ 1} : [Fed69, 1.4.5] If f ∈ Hom(V; V ) and dim V = m and t ∈ R, then m m det(idV +tf) = Q t tr(⋀kf) : k=0 [Fed69, 1.6.1] The Grassmannian of k dimensional vector subspaces of Rn is defined to be the set n G(n; k) = ξ ∈ ⋀kR ∶ ξ is simple~ ∼ ; where ξ ∼ η if and only if ξ = cη for some c ∈ R. n n n n ● Exercise. Let Ψ ∶ R → R × R be the diagonal map, i.e, Ψ(x) = (x; x) for x ∈ R ∗ ∗ ∗ and let p ∈ O (n; m), q ∈ O (n; n − k) be fixed and such that q ○ p = 0. For (g; h) ∈ O(k) × O(n − k) we define 'g;h ∈ O(n) to be the composition n Ψ n n p×q k n−k g×h k n−k ≃ n R ÐÐ→ R × R ÐÐÐ→ R × R ÐÐÐ→ R × R ÐÐÐ→ R : Next, we define the right action of (g; h) ∈ O(k) × O(n − k) on f ∈ O(n) by f ⋅ (g; h) = f ○ 'g;h : Show that under this action G(n; k) is homeomorphic with the quotient space, i.e., G(n; k) ≃ O(n)~O(k) × O(n − k) : ● Exercise. Consider the map n Rn n π ∶ {ξ ∈ ⋀kR ∶ ξ is simple} → 2 ; π(ξ) = {v ∈ R ∶ ξ ∧ v = 0} : Show that there exists a bijection j ∶ im π → G(n; k). n n Remark. The Hodge star (cf. [Fed69, 1.7.8]) operator ⋆ ∶ ⋀kR → ⋀n−kR gives rise to orthogonal complements under π, i.e., ⊥ π(ξ) = π(⋆ξ) : ● Exercise. Prove that G(n; m) is a smooth compact manifold of dimension m(n − m); cf. [Fed69, 3.2.28(2)(4)]. n Actually, G(n; m) can be isometrically embedded into the vectorspace B2 ⋀mR . Last update: November 6, 2017 Page 3 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński [All72, 2.3] With S ∈ G(n; m) we associate the n n orthogonal projection S♮ ∈ Hom(R ; R ) so that ∗ S♮ = S♮ ;S♮ ○ S♮ = S♮ ; im(S♮) = S: n n ● Exercise. If f ∈ Hom(R ; R ) and S ∈ G(n; k), then d 2 d 2 V Z⋀k((idRm +tf) ○ S♮)Z = V T⋀k((idRm +tf) ○ S♮)T = 2f ● S♮ : dt t=0 dt t=0 [All72, 8.9(3)] If S; T ∈ G(n; m), then ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ YS♮ − T♮Y = YS♮ ○ T♮Y = YT♮ ○ S♮Y = YS♮ ○ T♮ Y = YT♮ ○ S♮ Y = YS♮ − T♮ Y : n n n n [All72, 2.3(4)] If ! ∈ Hom(R ;R) and v ∈ R , then ! ⋅ v ∈ Hom(R ; R ) is given by (! ⋅ v)(u) = !(u)v and for S ∈ G(n; k) (! ⋅ v) ● S♮ = !(S♮(v)) = ⟨S♮v; !⟩ : Measures and measurable sets X ¯ [Fed69, 2.1.2] We say that φ measures X, if φ ∶ 2 → {t ∈ R ∶ 0 ≤ t ≤ ∞} and X φ(A) ≤ Q φ(B) whenever F ⊆ 2 is countable and A ⊆ F: B∈F A ⊆ X is said to be φ measurable if ∀T ⊆ X φ(T ) = φ(T ∩ A) + φ(T ∼ A) : [Fed69, 2.2.3] Let X be a topological space and φ measure X. We say that φ is Borel regular if all open sets in X are φ measurable and for each A ⊆ X there exists a Borel set B such that A ⊆ B and φ(A) = φ(B) : [Fed69, 2.2.5] Let X be a locally compact Hausdorff topological space and φ measure X. We say that φ is a Radon measure if all open sets are φ measurable and φ(K) < ∞ for K ⊆ X compact ; φ(V ) = sup{φ(K) ∶ K ⊆ V compact} for V ⊆ X open ; φ(A) = inf{φ(V ) ∶ A ⊆ V;V ⊆ X is open} for arbitrary A ⊆ X: n [Mat95, 14.15] For r > 0 let L(r) be the set of all maps f ∶ R → [0; ∞) such that spt(f) ⊆ B(0; r) and n Lip(f) ≤ 1. The space of all Radon measures over R equipped with the weak topology is a complete separable metric space. The metric is given by ∞ −1 d(φ, ) = Q 2 min{1;Fi(φ, )} ; where Fr(φ, ) = sup T∫ f dφ − ∫ f d T ∶ f ∈ L(r) : i=1 [All72, 2.6(2)] Let X be locally compact Hausdorff space. If G is a family of opens sets of X such that ⋃ G = X and B ∶ G → [0; ∞), then the set {φ ∶ φ is a Radon measure over X ; φ(U) ≤ B(U) for U ∈ G} Last update: November 6, 2017 Page 4 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński is (weakly) compact in the space of all Radon measures over X.