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 ϕ ∶ 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 ϕ ∈ ⋀ (V,Z) there exists a unique linear map ϕ˜ ∈ Hom(W, Z) such that ϕ = ϕ˜ ○ µ. µ V × ⋯ × V / W ∃!ϕ ˜ ∀ϕ  % Z We shall write W = ⋀kV and µ(v1, . . . , vk) = v1 ∧ ⋯ ∧ vk . k We shall frequently identify ϕ ∈ ⋀ (V,Z) with ϕ˜ ∈ Hom(⋀kV,Z). [Fed69, 1.3.2] If V = span{v1, . . . , vm}, then

⋀kV = span™vλ(1) ∧ ⋯ ∧ vλ(k) ∶ λ ∈ Λ(m, k)ž = span™vλ ∶ λ ∈ Λ(m, k)ž , 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 ω1, . . . , ωk 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 ωi(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 simplež~ ∼ ,

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 ϕg,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 ○ ϕg,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 ω ∈ Hom(R ,R) and v ∈ R , then ω ⋅ v ∈ Hom(R , R ) is given by (ω ⋅ v)(u) = ω(u)v and for S ∈ G(n, k) (ω ⋅ v) ● S♮ = ω(S♮(v)) = ⟨S♮v, ω⟩ .

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, ψ) = Q 2 min{1,Fi(φ, ψ)} , where Fr(φ, ψ) = sup ™T∫ f dφ − ∫ f dψT ∶ f ∈ L(r)ž . 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. If φi, φ are Radon measures and limi→∞ φi = φ, then φ U lim inf φ U for U X open , ( ) ≤ i→∞ ( ) ⊆ φ(K) ≥ lim sup φ(K) for K ⊆ X compact , i→∞ φ A lim φi A if Clos A is compact and φ Bdry A 0 . ( ) = i→∞ ( ) ( ) =

[Fed69, 2.10.2] Let Γ be the Euler function; see [Fed69, 3.2.13]. Assume X is a metric space. For m ∈ [0, ∞), δ > 0, and any A ⊆ X we set m −m m m ζ (A) = α(m)2 diam(A) , where α(m) = Γ(1~2) ~Γ((m + 2)~2) ,

m m G a countable family of subsets of X with Hδ (A) = inf œ Q ζ (S) ∶ ¡ . S∈G A ⊆ G and ∀S ∈ G diam(S) ≤ δ m The m dimensional Hausdorff measure H (A) of A ⊆ X is m m m H (A) = sup Hδ (A) = lim Hδ (A) . δ>0 δ↓0 m [Fed69, 2.10.33] Isodiametric inequality: If ∅ ≠ S ⊆ R , then m m −m m m L (S) = H (S) ≤ α(m)2 diam(S) = ζ (S) . n [Fed69, 4.1.4] Constancy theorem for distributions: If U ⊆ R is open, Y is a Banach space, T ∈ ′ D (U, Y ), A ⊆ U is connected, and

spt DjT ⊆ U ∼ A for j = 1, 2, . . . , n ,

then there exists a continuous linear function α ∶ Y → R such that n T (f) = α ○ f dL whenever f ∈ D(U, Y ) and spt f ⊆ A. SU

Approximate limits m n m m [Fed69, 2.9.12] Let A ⊆ R , f ∶ A → R , φ be a Radon measure over R , x ∈ R . φ({z ∈ B(x, r) ∶ Sf(z) − yS > ε}) φ ap lim f(z) = y ⇐⇒ ∀ε > 0 lim = 0 , z→x r↓0 φ(B(x, r)) φ({z ∈ B(x, r) ∶ f(z) > t}) φ ap lim sup f(z) = inf œt ∈ R ∶ lim = 0¡ , z→x r↓0 φ(B(x, r)) φ({z ∈ B(x, r) ∶ f(z) < t}) φ ap lim inf f(z) = sup œt ∈ R ∶ lim = 0¡ . z→x r↓0 φ(B(x, r))

Densities [Fed69, 2.10.19] Let φ be a Borel regular measure over a metric space X, m ∈ R, m ≥ 0, a ∈ X. We define ∗m −1 −m m −1 −m Θ (φ, a) = lim sup α(m) r φ(B(a, r)) , Θ∗ (φ, a) = lim inf α(m) r φ(B(a, r)) . r↓0 r↓0

m ∗m m If Θ∗ (φ, a) = Θ (φ, a), then we write Θ (φ, a) for the common value. Last update: November 6, 2017 Page 5 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński

∗m [Fed69, 2.10.19(1)] If A ⊆ X, t > 0, and Θ (φ, x) < t for all x ∈ A, then

m m φ(A) ≤ 2 tH (A) .

∗m [Fed69, 2.10.19(3)] If A ⊆ X, t > 0, and Θ (φ, x) > t for all x ∈ A, then for any open set V ⊆ X such that A ⊆ V m φ(V ) ≥ tH (A) . [Fed69, 2.10.19(4)] If A ⊆ X, φ(A) < ∞, and A is φ measurable, then

m m Θ (φ A, x) = 0 for H almost all x ∈ X ∼ A.

[Fed69, 2.10.19(2)(5)] If A ⊆ X, then

−m ∗m m m 2 ≤ Θ (H A, x) ≤ 1 for H almost all x ∈ A.

Tangent and normal vectors Let X be a normed vectorspace, φ a measure over X, a ∈ X, m a positive integer, S ⊆ X. [Fed69, 3.1.21] Tangent cone:

Tan(S, a) = {v ∈ X ∶ ∀ε > 0 ∃x ∈ S ∃r > 0 Sx − aS < ε and Sr(x − a) − vS < ε} , Tan(S, a) ∩ {v ∶ SvS = 1} =  Clos{(x − a)~Sx − aS ∶ a ≠ x ∈ S ∩ U(a, ε)} . ε>0

If the norm in X comes from a scalar product, define the normal cone

Nor(S, a) = {v ∈ X ∶ ∀τ ∈ Tan(S, a) v ● τ ≤ 0} .

[Fed69, 3.2.16] Approximate tangent cone:

m m Tan (φ, a) = {Tan(S, a) ∶ S ⊆ X, Θ (φ X ∼ S, a) = 0} .

If the norm in X comes from a scalar product, define the approximate normal cone

m m Nor (φ, a) = {v ∈ X ∶ ∀τ ∈ Tan (φ, a) v ● τ ≤ 0} .

For a ∈ X, v ∈ X, and ε > 0 define the cone

E(a, v, ε) = {x ∈ X ∶ ∃r > 0 Sr(x − a) − vS < ε} .

If the norm in X comes from a scalar product, v ∈ X, and 0 < ε < SvS, then b ∈ E(a, v, ε) if and only if 2 1~2 b − a v ε b ≠ a and ● > Œ1 − ‘ . Sb − aS SvS SvS2 Observe m ∗m v ∈ Tan (φ, a) ⇐⇒ ∀ε > 0 Θ (φ E(a, v, ε), a) > 0 .

Last update: November 6, 2017 Page 6 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński

Approximate differentiation Let X, Y be normed vectorspaces, φ be a measure over X, A ⊆ X, f ∶ A → Y , a ∈ X, m be a positive integer. [Fed69, 3.2.16] We say that f is (φ, m) approximately differentiable at a if there exists an open neigh- bourhood U of a in X and a function g ∶ U → Y such that m Dg(a) exists and Θ (φ {x ∈ A ∶ f(x) ≠ g(x)}, a) = 0 . We then define m (φ, m) ap Df(a) = Dg(a)STanm(φ,a) ∈ Hom(Tan (φ, a),Y ) . Observe that (φ, m) ap Df(a) exists if and only if there exist y ∈ Y and continuous L ∈ Hom(X,Y ) such that for each ε > 0 m Θ (φ X ∼{x ∶ Sf(x) − y − L(x − a)S ≤ εSx − aS}, a) = 0 .

m n Jacobians Assume A ⊆ R and f ∶ A → R . m n [Fed69, 3.2.1] If a ∈ A and Df(a) ∈ Hom(R , R ) exists, then the k-dimensional Jacobian Jkf(a) ∈ R of f at a is defined by Jkf(a) = Y⋀kDf(a)Y . In case k = min{m, n}, we have ∗ 1~2 ∗ 1~2 Jkf(a) = S⋀kDf(a)S = tr‰⋀k(Df(a) ○ Df(a))Ž = tr‰⋀k(Df(a) ○ Df(a) )Ž . In particular, if k = m ≤ n, then ∗ 1~2 Jkf(a) = det(Df(a) ○ Df(a)) and if k = n ≤ m, then ∗ 1~2 Jkf(a) = det(Df(a) ○ Df(a) ) . m m m n If φ measures R , m is a positive integer, a ∈ R , and (φ, m) ap Df(a) ∈ Hom(R , R ) exists, then the (φ, m) approximate k-dimensional Jacobian (φ, m) ap Jkf(a) ∈ R of f at a is defined by (φ, m) ap Jkf(a) = Y⋀k(φ, m) ap Df(a)Y .

Lebesgue integral Assume φ measures X. [Fed69, 2.4.1] We say that u is a φ step function if u is φ measurable, im(u) is a countable subset of R, and −1 ¯ Q y φ(u {y}) ∈ R . y∈im(u) ¯ [Fed69, 2.4.2] Let f ∶ X → R. Set

∗ ⎧ ⎫ ⎪ −1 u is a φ step function and ⎪ S f dφ = inf ⎨ Q y φ(u {y}) ∶ ⎬ , ⎪y∈im(u) u(x) ≥ f(x) for φ almost all x⎪ ⎩⎪ ⎭⎪ ⎪⎧ u is a φ step function and ⎪⎫ f dφ sup ⎪ y φ u−1 y ⎪ . S∗ = ⎨ Q ( { }) ∶ ⎬ ⎪y∈im(u) u(x) ≤ f(x) for φ almost all x⎪ ⎩⎪ ⎭⎪ ∗ We say that f is φ integrable if ∫∗ f dφ = ∫ f dφ and then we write ∫ f dφ for the common value. We say that f is φ summable if S ∫ f dφS < ∞. Last update: November 6, 2017 Page 7 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński

n n [Fed69, 2.9.1] If φ, ψ are Radon measures over R and x ∈ R , we define D φ, ψ, x lim φ B x, r ψ B x, r . ( ) = r↓0 ( ( ))~ ( ( ))

[Fed69, 2.9.5] 0 ≤ D(φ, ψ, x) < ∞ for ψ almost all x. n [Fed69, 2.9.7] If A ⊆ R is ψ measurable, then

D(φ, ψ, x) dψ(x) ≤ φ(A) , SA with equality if and only if φ is absolutely continuous with respect to ψ. [Fed69, 2.9.19] If ∞ ≤ a < b ≤ ∞ and f ∶ (a, b) → R is monotone (or, more generally, a function of 1 bounded variation), then f is differentiable at L almost all t ∈ (a, b) and

b ′ 1 V f dL V ≤ Sf(b) − f(a)S . Sa

[Fed69, 2.5.12] Theorem. Let X be a locally compact separable metric space, E a separable normed vectorspace, T ∶ K (X,E) → R be linear and such that

sup{T (ω) ∶ ω ∈ K (X,E) , spt ω ⊆ K, SωS ≤ 1} < ∞ whenever K ⊆ X is compact .

Define

φ(U) = sup {T (ω) ∶ ω ∈ K (X,E) , SωS ≤ 1 , spt ω ⊆ U} whenever U ⊆ X is open , φ(A) = inf {φ(U) ∶ A ⊆ U,U ⊆ X is open} for arbitrary A ⊆ X.

∗ Then φ is a Radon measure over X and there exists a φ measurable map k ∶ X → E such that Yk(x)Y = 1 for φ almost all x and

T (ω) = S ⟨ω(x), k(x)⟩ dφ(x) for ω ∈ K (X,E) .

See also: [Sim83, 4.1]

Last update: November 6, 2017 Page 8 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński

Area and co-area formulas. Rectifiability. m n [Fed69, 3.2.3] Theorem. Suppose f ∶ R → R , and Lip(f) < ∞, and m ≤ n. m m (a) If A ⊆ R is L measurable, then m m Jmf dL = N(fSA, y) dH (y) . SA SRn m m (b) If u ∶ R → R is L integrable, then m m u x Jmf x dL x u x dH y . S ( ) ( ) ( ) = S n Q ( ) ( ) R x∈f −1{y} m n m [Fed69, 3.2.5] Theorem. Suppose f ∶ R → R , and Lip(f) < ∞, and m ≤ n, and A ⊆ R is m m ¯ L measurable, and g ∶ R → R. Then m m g(f(x))Jmf(x) dL (x) = g(y)N(fSA, y) dH (y) SA SRn given (a) either g is H m measurable m n (b) or N(fSA, y) < ∞ for H almost all y ∈ R m (c) or 1A ⋅ (g ○ f) ⋅ Jmf is L measurable. m n [Fed69, 3.2.11-12] Theorem. Suppose f ∶ R → R , and Lip(f) < ∞, and m > n. m m (a) If A ⊆ R is L measurable, then m m−n −1 n Jnf dL = H (f {y}) dL (y) . SA SRn m ¯ m (b) If u ∶ R → R is L integrable, then m m−n n u(x)Jnf(x) dL (x) = u(x) dH (x) dL (y) . S SRn Sf −1{y} n n [Fed69, 3.2.14] Definition. Let E ⊆ R , m be a positive integer, φ measures R . m n (a) E is m rectifiable if there exists ϕ ∶ R → R with Lip(ϕ) < ∞ and such that m E = ϕ[A] for some bounded set A ⊆ R ; (b) E is countably m rectifiable if is a union of countably many m rectifiable sets; n (c) E is countably (φ, m) rectifiable if there exists a countably m rectifiable set A ⊆ R such that φ(E ∼ A) = 0; (d) E is (φ, m) rectifiable if E is countably (φ, m) rectifiable and φ(E) < ∞. m n (e) E is purely (φ, m) unrectifiable if φ(E ∩ im ϕ) = 0 for all ϕ ∶ R → R with Lip(ϕ) < ∞. n m [Fed69, 3.2.29] Theorem. A set W ∈ R is countably (H , m) rectifiable if and only if there ex- ists a countable family F of m dimensional submanifolds of Rn of class C 1 such that m H (W ∼ ⋃ F ) = 0. n m m [Fed69, 3.2.18] Lemma. Assume W ⊆ R is (H , m) rectifiable and H measurable. Then for each m m n λ ∈ (1, ∞), there exist compact subsets K1,K2,... of R and maps ψ1, ψ2,... ∶ R → R such that m ∞ {ψi[Ki] ∶ i = 1, 2,...} is disjointed , H (W ∼ ⋃i=1ψi[Ki]) = 0 , −1 Lip(ψi) ≤ λ , ψiSKi is injective , Lip((ψiSKi ) ) ≤ λ , −1 m λ SvS ≤ SDψi(a)vS ≤ λSvS for a ∈ Ki , v ∈ R . Last update: November 6, 2017 Page 9 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński

n m m [Fed69, 3.2.19] Theorem. Assume W ⊆ R is (H , m) rectifiable and H measurable. Then for m H almost all w ∈ W

m m m m Θ (H W, w) = 1 and Tan (H W, w) ∈ G(n, m) .

ν Moreover, if f ∶ W → R and Lip(f) < ∞, then

m m m ν (H W, m) ap Df(w) ∶ Tan (H W, w) → R

m exists for H almost all w ∈ W . n m m [Fed69, 3.2.20] Corollary. Let W ⊆ R be (H , m) rectifiable and H measurable. Assume m ≤ ν, ν and f ∶ W → R , and Lip(f) < ∞. Then

m m (g ○ f)Jmf dH = g(z)N(f, z) dH (z) SW SRν ν ¯ for any g ∶ R → R. n m m m [Mat75, Pre87] Theorem. If W ⊆ R and Θ (H W, w) = 1 for H almost all w ∈ W , then W is m countably (H , m) rectifiable. n m m [Fed69, 3.2.22] Theorem. Let m ≥ µ, and W ⊆ R be (H , m) rectifiable and H measurable, and ν µ µ Z ⊆ R be (H , µ) rectifiable and H measurable, and f ∶ W → Z, and Lip(f) < ∞. m For brevity let us write “ap” for “(H W, m) ap”. m (a) For H almost all w ∈ W , either ap Jµf(w) = 0 or

µ µ im ap Df(w) = Tan (H Z, f(w)) ∈ G(ν, µ) .

−1 m−µ m−µ µ (b) The levelset f {z} is (H , m−µ) rectifiable and H measurable for H al- most all z ∈ Z. m ¯ (c) For any (H W ) integrable function g ∶ W → R

m m−µ µ g ⋅ ap Jµf dH = g dH dH (z) . SW SZ Sf −1{z}

n ν µ [Fed69, 3.2.23] Theorem. Assume W ⊆ R is m rectifiable and Borel, and Z ⊆ R is (H , µ) rectifiable n ν m+µ and Borel. Then W × Z ⊆ R × R is (H , m + µ) rectifiable and

m+µ m µ H (W × Z) = (H W ) × (H Z) .

n ν m µ [Fed69, 3.2.24] Beware, there exist sets W ⊆ R and Z ⊆ R with H (W ) = 0 and H (Z) = 0 but m+µ m+µ m µ H (W × Z) = ∞. In particular, H (W × Z) ≠ (H W ) × (H Z)!

Last update: November 6, 2017 Page 10 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński

n BV, Caccioppoli sets, and the Gauss-Green theorem. Let U ⊆ R be open. 1 [EG92, 5.1] Definition. A function f ∈ L (U) has bounded variation in U if

n 1 n YDfY(U) = sup ›S f div ϕ dL ∶ ϕ ∈ Cc (U, R ) , SϕS ≤ 1 < ∞ .

We define

1 BV (U) = {f ∈ L (U) ∶ YDfY(U) < ∞} and YfYBV (U) = YfYL1(U) + YDfY(U) .

1 Definition. f ∈ L (U) has locally bounded variation in U if f ∈ BV (V ) for all open sets V ⊆ U such that Clos V ⊆ U is compact. We write f ∈ BVloc(U). n n Definition. An L measurable set E ⊆ R has finite perimeter in U if 1E ∈ BV (U). 1 Definition. E has locally finite perimeter in U if E ∈ BVloc(U). n Theorem. f ∈ BV (U) if and only if there exists a Radon measure µ over R and n a µ measurable function σ ∶ U → R satisfying Sσ(x)S = 1 for µ almost all x and

n 1 n f div ϕ dL = − ϕ ● σ dµ for ϕ ∈ Cc (U, R ) . SU SU Notation. (a) If f ∈ BVloc(U), then we write YDfY = µ and ∇f for the density of the abso- lutely continuous part of the vector-valued Radon measure µ σ with respect to the Lebesgue measure L n. n (b) If E ⊆ R has locally finite perimeter in U, then we write Y∂EY = YD1EY and νE = −σ. 1,1 1,1 [EG92, 5.1, Ex.1] Remark. We have Wloc (U) ⊆ BVloc(U). Moreover, for f ∈ Wloc (U) and any A ⊆ U

n YDfY(A) = S grad fS dL and ∇f = grad f . SA n [EG92, 5.1, Ex.2] Remark. If E ⊆ R is open and the topological boundary Bdry E is a smooth hyper- n n−1 surface in R such that H (Bdry E ∩ K) < ∞ for all compact K ⊆ U, then E has n−1 locally finite perimeter in U. Moreover, if H (Bdry E) < ∞, then

n−1 Y∂EY = H Bdry E and νE is the outer unit normal to Bdry E.

1 [EG92, 5.2.1] Theorem. If fi ∈ BV (U) and fi → f in Lloc(U), then

Df U lim inf Dfi U . Y Y( ) ≤ i→∞ Y Y( )

[EG92, 5.2.2] Theorem. Assume f ∈ BV (U). Then there exist functions fi ∈ BV (U) ∩ E (U, R) such that

1 fi → f in L (U) and YDfiY(U) → YDfY(U) as i → ∞ n and L grad fi → YDfY σ weakly as vector-valued Radon measures .

[EG92, 5.2.3] Theorem. Assume U is open and bounded in Rn, Bdry U is a Lipschitz manifold, fi ∈ BV (U) satisfies sup{YfiYBV (U) ∶ i = 1, 2,...} < ∞. Then there exists a subsequence 1 fkj and a function f ∈ BV (U) such that fkj → f in L (U). Last update: November 6, 2017 Page 11 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński

[EG92, 5.5] Remark. If f ∶ U → R is Lipschitsz, then the co-area formula gives n n−1 −1 1 S S grad fS dL = S H (f {t}) dL (t) . 1 Theorem. Let f ∈ L (U) and define for t ∈ R Et = {x ∈ U ∶ f(x) > t} . 1 (a) If f ∈ BV (U), then Et has finite perimeter in U for L almost all t. (b) If f ∈ BV (U), then 1 YDfY(U) = S Y∂EtY(U)L (t) . 1 (c) If ∫ Y∂EtY(U)L (t) < ∞, then f ∈ BV (U). n [EG92, 5.6.2] Theorem. Let E be bounded and of finite perimeter in R . There exists C = C(n) > 0 such that n 1−1~n n (a) L (E) ≤ CY∂EY(R ), n n 1−1~n n (b) min{L (B(x, r) ∩ E), L (B(x, r) ∼ E)} ≤ CY∂EY(U(x, r)) for x ∈ R , r ∈ (0, ∞). n n [EG92, 5.7.1] Definition. Assume E has locally finite perimeter in R and x ∈ R . We say that x belongs to the reduced boundary ∂∗E of E if (a) Y∂EY(B(x, r)) > 0 for r > 0, lim ∂E B x, r −1 ν d ∂E ν x (b) r↓0 Y Y( ( )) ∫B(x,r) E Y Y = E( ), (c) SνE(x)S = 1. [EG92, 5.7.3] Theorem. Assume E has locally finite perimeter in Rn. ∗ n−1 (a) ∂ E is countably (H , n − 1) rectifiable. n−1 ∗ n (b) H (∂ E ∩ K) < ∞ for any compact set K ⊆ R . n−1 n−1 ∗ n−1 ∗ (c) νE(x) ∈ Nor (H ∂ E, x) for H almost all x ∈ ∂ E. n−1 ∗ (d) Y∂EY = H ∂ E. n n [EG92, 5.8] Definition. Assume E has locally finite perimeter in R and x ∈ R . We say that x belongs to the measure theoretic boundary ∂∗E of E if ∗n n ∗n n n Θ (L E, x) > 0 and Θ (L (R ∼ E), x) > 0 . ∗ n−1 ∗ Lemma. ∂ E ⊆ ∂∗E and H (∂∗E ∼ ∂ E) = 0. Theorem. Assume E has locally finite perimeter in Rn. Then n n−1 1 n n div ϕ dL = ϕ ● νE dH for ϕ ∈ Cc (R , R ) . SE S∂∗E n n n [EG92, 5.11] Theorem. Let E ⊆ R be L measurable. Then E has locally finite perimeter in R n−1 n if and only if H (∂∗E ∩ K) < ∞ for all compact sets K ⊆ R . n n n [EG92, 6.1.1] Theorem. Assume f ∈ BVloc(R ). Then for L almost all x ∈ R 1−1~n 1 −1 −n n~(n−1) n lim ‹α(n) r Sf(y) − f(x) − ∇f(x) ● (x − y)S dL (y) = 0 . r↓0 r SB(x,r) n n [EG92, 6.1.3] Theorem. Assume f ∈ BVloc(R ). Then f is (L , n) approximately differentiable L n almost everywhere. Moreover, n n n n (L , n) ap Df(x)u = ∇f(x) ● u for L almost all x ∈ R and all u ∈ R .

Last update: November 6, 2017 Page 12 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński

n Varifolds – definitions. Let U ⊆ R be open and M ⊆ U be a smooth m dimensional n submanifold (possibly open) such that the inclusion map i ∶ M ↪ R is proper. [All72, 2.5] Definition. ∞ n • tangent vector fields: X (M) = {g ∈ Cc (M, R ) ∶ ∀x ∈ M g(x) ∈ Tan(M, x)}; ⊥ ∞ n • normal vector fields: X (M) = {g ∈ Cc (M, R ) ∶ ∀x ∈ M g(x) ∈ Nor(M, x)}; ∞ n • tangent and normal parts of a vectorfield: if g ∈ Cc (M, R ), then Tan(M, g) ∈ ⊥ X (M) and Nor(M, g) ∈ X (M) are such that g = Tan(M, g) + Nor(M, g); • Gk(M) = {(x, S) ∶ x ∈ M,S ∈ G(n, k) ,S ⊆ Tan(M, x)}; • the second fundamental form: b(M, a) ∶ Tan(M, a) × Tan(M, a) → Nor(M, a) a symmetric bilinear mapping such that

⊥ Dg(a)w ● v = −b(M, a)(v, w) ● g(a) for v, w ∈ Tan(M, a) and g ∈ X (M) ;

• the mean curvature vector: h(M, a) ∈ Nor(M, a) is characterized by

⊥ (Dg(a) ○ Tan(M, a)♮) ● Tan(M, a)♮ = −g(a) ● h(M, a) for g ∈ X (M) ;

• for (a, S) ∈ Gk(M) the vector h(M, a, S) ∈ Nor(M, a) is characterized by

⊥ (Dg(a) ○ Tan(M, a)♮) ● S♮ = −g(a) ● h(M, a, S) for g ∈ X (M) .

[All72, 3.1] Definition. A Radon measure V over Gk(M) is called a k dimensional varifold in M. The weakly topologised space of k dimensional varifolds in M is denoted Vk(M). For any V ∈ Vk(M) we define the weight measure YV Y over M by requiring

YV Y(B) = V ({(x, S) ∈ Gk(M) ∶ x ∈ B}) for B ⊆ M Borel .

′ [All72, 3.2] Definition. If F ∶ M → M is a smooth map between smooth manifolds and V ∈ Vk(M), ′ then we define F#V ∈ Vk(M ) by

′ F#V (α) = S α(F (x), DF (x)[S])Y⋀kDF (x) ○ S♮Y dV (x, S) for α ∈ K (Gk(M )) ,

with the understanding that α(F (x), DF (x)[S])Y⋀kDF (x) ○ S♮Y equals zero whenever ⋀kDF (x) ○ S♮ = 0. Remark. Observe k Yµr#V Y = r µr#YV Y .

[All72, 3.3] Definition. (Varifold disintegration; cf. [AFP00, §2.5]) For V ∈ Vk(M) we define for x ∈ M and β ∈ K (G(n, k))

(x) −1 V (β) = lim Yi#V Y(B(x, r)) β(S) d(i#V )(y, S) . r↓0 SB(x,r)×G(n,k)

n [All72, 3.4] Definition. Let V ∈ Vk(M), a ∈ M, and j ∶ Tan(M, a) ↪ R be the inclusion map.

VarTan V, a C Vk Tan M, a j#C lim µr τ −a i #V for some rj . ( ) = › ∈ ( ( )) ∶ = j→∞( j ○ ○ ) ↑ ∞

Last update: November 6, 2017 Page 13 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński

n k k [All72, 3.5] Definition. If E ⊆ R is countably (H , k) rectifiable and H (E ∩ K) < ∞ for K ⊆ U compact, then define vk(E) ∈ Vk(U) by

k k k vk(E)(α) = α(x, Tan (H E, x)) dH (x) for α ∈ K (Gk(U)) . SE

Definition. We say that V ∈ Vk(M) is a rectifiable varifold if there exist countably m (H , m) rectifiable sets Ei ⊆ M and constants ci ∈ (0, ∞) such that ∞ V = Q civk(Ei) . i=1

If all ci can be taken to be integers, then we say that V is an integral varifold. The spaces of all k dimensional rectifiable and integral varifolds in M are denoted by

RVk(M) and IVk(M) .

Theorem. Let V ∈ Vk(M). Then V ∈ RVk(M) if and only if for YV Y almost all a

m (a) k Θ (i#YV Y, a) ∈ (0, ∞) and V (β) = β(Tan (i#YV Y, a)) for β ∈ K (G(n, k)) .

m Moreover, V ∈ IVk(M) if and only if V ∈ RVk(M) and Θ (i#YV Y, a) is a non-negative integer for YV Y almost all a. [All72, 4.2] Definition. Let V ∈ Vk(M). Define δV ∶ X (M) → R the first variation of V by

δV (g) = S (Dg(x) ○ S♮) ● S♮ dV (x, S) for g ∈ X (M) .

Definition. The total variation measure YδV Y is given by

YδV Y(G) = sup {δV (g) ∶ g ∈ X (M) , spt g ⊆ G, SgS ≤ 1} for G ⊆ M open , YδV Y(A) = inf {YδV Y(G) ∶ A ⊆ G,G ⊆ M open} for arbitrary A ⊆ M.

Definition. If δV = 0, we say that V is stationary. If G ⊆ M is open and YδV Y(G) = 0, we say that V is stationary in G. [All72, 4.3] Definition. Assume YδV Y is a Radon measure. Then there exists a YδV Y measurable function η(V, ⋅) such that for YδV Y almost all x there holds η(V, x) ∈ Tan(M, s) and

δV (g) = S g(x) ● η(V, x) dYδV Y(x) for g ∈ X (M) .

Setting h(V, x) = −D(YδV Y, YV Y, x)η(V, x) we obtain a YV Y measurable function such that

δV (g) = − S g(x) ● h(V, x) dYV Y(x) + S g(x) ● η(V, x) dYδV Ysing(x) for g ∈ X (M) ,

where YδV Ysing denotes the singular part of YδV Y with respect to YV Y. We call h(V, x) the generalized mean curvature vector of V at x.

Last update: November 6, 2017 Page 14 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński

n Varifolds – examples and basic facts. Let U ⊆ R be open and M ⊆ U be a smooth n m dimensional submanifold (possibly open) such that the inclusion map i ∶ M ↪ R is proper. [All72, 4.4] Remark. If V ∈ Vk(M) and g ∈ X (U), then

δ(i#V )(g) = δV (Tan(M, g)) − S Nor(M, g)(x) ● h(M, x, S) dV (x, S) .

m [All72, 4.5] Lemma. Let W ⊆ U be open, Y ⊆ R be open, ϕ ∶ Y → W and ψ ∶ W → Y be smooth and such that ψ ○ ϕ = idY and W ∩ M = W ∩ im ϕ, V ∈ Vm(M). Then

δV (g) = δ(ψ#V )(Y⋀mDϕY⟨g ○ ϕ, Dψ ○ ϕ⟩) for g ∈ X (W ∩ M) , −1 m Dβ(y)v dYψ#V Y(y) = δV ‰(Y mDϕY β ⋅ Dϕ(⋅)v) ○ ψŽ for v ∈ R and β ∈ D(Y, R) . SY ⋀

[All72, 4.6] Theorem. Assume M is connected, dim M = m, V ∈ Vm(U), spt YV Y ⊆ M, YδV Y is a Radon measure, and

δV (g) = 0 for g ∈ X (M) with Nor(M, g) = 0 .

Then there exists a constant C > 0 such that

m m V = Cvm(M) and C = YV Y(A)~H (A) for any A ⊆ M with H (A) ∈ (0, ∞) .

[All72, 4.7] Example. If E ⊆ M is a set of locally finite perimeter in M, then vm(E) ∈ Vm(M) and

m−1 δvm(E)(g) = g(x) ● νE(x) dH (x) for g ∈ X (M) . S∂∗E

n [All72, 4.8] Example. Let 0 < k < n and T ∈ G(n, k). Set V (A) = H ({x ∶ (x, T ) ∈ A}) for n A ⊆ R × G(n, k). Then

n n k n V ∈ Vk(R ) , δV = 0 , YV Y = H , Θ (YV Y, a) = 0 for a ∈ R .

n ● Exercise. Let 0 < k < n, and Σ be a smooth k-dimensional submanifold of R with 1 smooth boundary, and θ ∶ Σ → (0, ∞) be of class C . Define

k n V (α) = S α(x, Tan(Σ, x))θ(x) dH (x) for α ∈ K (R × G(n, k)) .

n For g ∈ X (R ) we have

k δV (g) = − g(x) ● (h(Σ, x) + Tan(Σ, x)♮‰grad(log ○θ)(x)Žθ(x) dH (x) SΣ k−1 + g(x) ● νΣ(x)θ(x) dH (x) , S∂Σ

where νΣ(x) is the unit normal vector to Σ at x ∈ ∂Σ. In particular,

k YδV Ysing = θH ∂Σ , η(V, x) = νΣ(x) for x ∈ ∂Σ , h(V, x) = h(Σ, x) + Tan(Σ, x)♮‰grad(log ○θ)(x)Ž for x ∈ Σ .

Last update: November 6, 2017 Page 15 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński

[All72, 4.10] Lemma. Assume r ∈ R, V ∈ Vk(U), YδV Y is a Radon measure, f ∶ W → R is continuous, −1 g ∈ X (U), f is smooth in a neighborhood of spt YV Y ∩ f {r} ∩ spt g. Then

(δV {x ∶ f(x) > r})(g) = δ‰V {(x, S) ∶ f(x) > r}(g)Ž(g) 1 + lim S♮(g(x)) ● grad f(x) dV (x, S) . h↓0 h S{(x,S)∶r

Remark. Set Er = {x ∈ U ∶ f(x) > r}. In the language of [Men16, §5] one could write 1 V ∂Er(g) = lim S♮(g(x)) ● grad f(x) dV (x, S) . h↓0 h S{(x,S)∶r

Theorem. Assume V ∈ Vk(U), YδV Y is a Radon measure, −∞ ≤ a < b ≤ ∞, f ∶ W → R −1 1 is continuous and smooth in a neighborhood of spt YV Y ∩ f (a, b). Then for L almost all r ∈ (a, b) the measure Yδ(V {(x, S) ∶ f(x) > r})Y is a Radon measure and

b 1 Yδ(V {(x, S) ∶ f(x) > r})Y(B) dL (r) Sa b 1 ≤ SS♮(grad f(x))S dV (x, S) + YδV Y(B ∩ {x ∶ f(x) > r}) dL (r) SB∩f −1(a,b)×G(n,k) Sa

for any Borel set B ⊆ U. n [All72, 4.12] Remark. Let V ∈ Vk(R ) and r ∈ (0, ∞).

k−1 Yδ(µr#V )Y = r µr#YδV Y .

k−1 Remark. If Θ (YδV Y, a) = 0, then all members of VarTan(V, a) are stationary.

Last update: November 6, 2017 Page 16 of 17 GMT – Varifolds Cheat-sheet Sławomir Kolasiński

References

[AFP00] Luigi Ambrosio, Nicola Fusco, and Diego Pallara. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.

[All72] William K. Allard. On the first variation of a varifold. Ann. of Math. (2), 95:417–491, 1972.

[EG92] Lawrence C. Evans and Ronald F. Gariepy. Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.

[Fed69] Herbert Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.

[Mat75] Pertti Mattila. Hausdorff m regular and rectifiable sets in n-space. Trans. Am. Math. Soc., 205:263–274, 1975. doi:10.2307/1997203.

[Mat95] Pertti Mattila. Geometry of sets and measures in Euclidean spaces, volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cam- bridge, 1995. Fractals and rectifiability. URL: http://dx.doi.org/10.1017/ CBO9780511623813, doi:10.1017/CBO9780511623813.

[Men16] Ulrich Menne. Weakly differentiable functions on varifolds. Indiana Univ. Math. J., 65(3):977–1088, 2016. URL: http://dx.doi.org/10.1512/iumj.2016.65.5829, doi:10.1512/iumj.2016.65.5829.

[Pre87] David Preiss. Geometry of measures in Rspn ∶ Distribution, rectifiability, and densi- ties. Ann. Math. (2), 125:537–643, 1987. doi:10.2307/1971410.

[Sim83] Leon Simon. Lectures on geometric measure theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University. Australian Na- tional University, Centre for Mathematical Analysis, Canberra, 1983.

Sławomir Kolasiński Instytut Matematyki, Uniwersytet Warszawski ul. Banacha 2, 02-097 Warszawa, Poland [email protected]

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