MAnET MID-TERM REVIEW MEETING Minimal cones and calibrations - PowerPoint PPT Presentation

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 R n with locally finite d-dimensional Hausdorff measure H d . Admissible competitors Let U ⊂ R n 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: H d ( E \ F ) ≤ H d ( F \ E ) . Cavallotto Edoardo Minimal cones and calibrations 2/11

Almgren minimisers Properties If E is an Almgren minimiser then: E is rectifiable; E is Ahlfors regular; E is uniformly rectifiable; there exists an H d -negligible N ⊂ R n such that E \ N is a C 1 ,α d -dimensional submanifold of R n ; ∀ x ∈ E , ∀ µ ∈ Tan d ( H d � E , x ) is supported on a minimal cone. Remark Let E ⊂ R n be an Almgren minimiser: E is an Almgren minimiser as a subset of R m with m > n because since projections do not increase the Hausdorff area; let k > 0, by a slicing argument E × R k is an Almgren minimiser in R n + k . Cavallotto Edoardo Minimal cones and calibrations 3/11

Minimal Cones R 2 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 . R 3 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 . Cavallotto Edoardo Minimal cones and calibrations 4/11

Minimal Cones Higher dimensions cone over sk n − 2 ( Q n ) for n ≥ 4 [Brakke 1991]; cone over sk n − 2 (∆ n ) [Morgan 1994]; for any d , m ≥ 2 there exists θ ( m , d ) ∈ ( 0 , π/ 2 ) such that, given P 1 , . . . , P m d-planes in R md , their union is an Almgren minimiser if all the characteristic angles are greater than θ ( m , d ) [Liang 2013]; Y × Y ⊂ R 4 [Liang 2014]. Cavallotto Edoardo Minimal cones and calibrations 5/11

Sliding Boundary Sliding deformation Let U ⊂ R n be open and Γ i ⊂ U , 1 ≤ i ≤ I , be a finite collection of closed sets. A sliding deformation with respect to { Γ i } 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 ) ∈ Γ i ∀ t if x ∈ Γ i . Sliding boundary minimisers Given Γ := ∪ i Γ i and 0 ≤ α ≤ 1 we define a new cost functional c ( E ) := H d ( E \ Γ) + α H d ( 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 R 2 + , 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 R 3 + , the sliding boundary Γ := { z = 0 } is in blue. Cavallotto Edoardo Minimal cones and calibrations 8/11

Sliding minimal cones Two-dimensional cones in R 3 + , 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 Y β , is minimal if cos β = α 2 3 , the √ right cone is not minimal. Cavallotto Edoardo Minimal cones and calibrations 9/11

Sliding minimal cones Let us set T 2 + as the cone over sk 1 (∆ 3 ) ∩ R 3 + , then T + is a sliding � 2 minimal cone if α ≥ 3 . In general, let T n − 1 be the cone over sk n − 1 (∆ n ) ∩ R n + , then T + is + � 1 n + 1 a sliding minimal cone if α ≥ n . √ 2 Cavallotto Edoardo Minimal cones and calibrations 10/11

KIITOS MIELENKIINNOSTANNE THANK YOU FOR YOUR ATTENTION Cavallotto Edoardo Minimal cones and calibrations 11/11