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MAnET MID-TERM REVIEW MEETING Minimal cones and calibrations Cavallotto Edoardo

December 9, 2015 Helsinki

Cavallotto Edoardo Minimal cones and calibrations 1/11

Almgren minimisers

Let us fix 0 < d < n, E will be a subset of Rn with locally finite d-dimensional Hausdorff measure Hd. Admissible competitors Let U ⊂ Rn be open. F is an admissible competitor for E on U if there exists a continuous function φ : [0, 1] × U → U such that: φ(0, ·) = Id; ∃K ⊂⊂ U such that ∀t φ(t, ·) = Id in K c; φ(1, ·) is Lipschitz and F = φ(1, E). We call φ an admissible deformation. Almgren minimiser E is an Almgren minimiser if for every admissible competitor F we have: Hd(E \ F) ≤ Hd(F \ E).

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Almgren minimisers

Properties If E is an Almgren minimiser then: E is rectifiable; E is Ahlfors regular; E is uniformly rectifiable; there exists an Hd-negligible N ⊂ Rn such that E \ N is a C 1,α d-dimensional submanifold of Rn; ∀x ∈ E, ∀µ ∈ Tand(HdE, x) is supported on a minimal cone. Remark Let E ⊂ Rn be an Almgren minimiser: E is an Almgren minimiser as a subset of Rm with m > n because since projections do not increase the Hausdorff area; let k > 0, by a slicing argument E × Rk is an Almgren minimiser in Rn+k.

Cavallotto Edoardo Minimal cones and calibrations 3/11

Minimal Cones

R2 There are only two types of minimal cones: the straight line; and three half lines meeting with angle of 120◦, which we will denote as Y . R3 There are no new 1-dimensional cones. The 2-dimensional minimal cones are of three kinds: planes; Y := Y × R; the cone over the edges of a regular tetrahedron T.

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Minimal Cones

Higher dimensions cone over skn−2(Qn) for n ≥ 4 [Brakke 1991]; cone over skn−2(∆n) [Morgan 1994]; for any d, m ≥ 2 there exists θ(m, d) ∈ (0, π/2) such that, given P1, . . . , Pm d-planes in Rmd, their union is an Almgren minimiser if all the characteristic angles are greater than θ(m, d) [Liang 2013]; Y × Y ⊂ R4 [Liang 2014].

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Sliding Boundary

Sliding deformation Let U ⊂ Rn be open and Γi ⊂ U, 1 ≤ i ≤ I, be a finite collection

• f closed sets. A sliding deformation with respect to {Γi}i is a

continuous function φ : [0, 1] × U → U such that: φ(0, ·) = Id; ∃K ⊂⊂ U such that ∀t φ(t, ·) = Id in K c; φ(1, ·) is Lipschitz and F = φ(1, E); φ(t, x) ∈ Γi ∀t if x ∈ Γi. Sliding boundary minimisers Given Γ := ∪iΓi and 0 ≤ α ≤ 1 we define a new cost functional c(E) := Hd(E \ Γ) + αHd(E ∩ Γ) and we say that E is a sliding boundary minimiser if for any competitor F obtained as image of a sliding deformation we have c(E \ F) ≤ c(F \ E).

Cavallotto Edoardo Minimal cones and calibrations 6/11

Sliding minimal cones

One-dimensional minimal cones in R2

+, the sliding boundary

Γ := {y = 0} is in blue.

Cavallotto Edoardo Minimal cones and calibrations 7/11

Sliding minimal cones

One-dimensional minimal cones in R3

+, the sliding boundary

Γ := {z = 0} is in blue.

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Sliding minimal cones

Two-dimensional cones in R3

+, the sliding boundary Γ := {z = 0} is

in blue, the intersection between Γ and the cone is in grey. The left cone, which we call Yβ, is minimal if cos β = α 2

√ 3, the

right cone is not minimal.

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Sliding minimal cones

Let us set T2

+ as the cone over sk1(∆3) ∩ R3 +, then T+ is a sliding

minimal cone if α ≥

• 2

3.

In general, let Tn−1

+

be the cone over skn−1(∆n) ∩ Rn

+, then T+ is

a sliding minimal cone if α ≥

1 √ 2

• n+1

n .

Cavallotto Edoardo Minimal cones and calibrations 10/11

KIITOS MIELENKIINNOSTANNE THANK YOU FOR YOUR ATTENTION

Cavallotto Edoardo Minimal cones and calibrations 11/11