One dimensional curvatures Higher dimensional Menger type curvatures Higher dimensional Menger curvature as a tool for proving regularity of sets Sławomir Kolasi´ nski Institute of Mathematics University of Warsaw November 30, 2011 2nd European Young and Mobile Workshop Universidad de Granada Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves Outline One dimensional curvatures 1 Curve thickness Integral curvatures for curves Higher dimensional Menger type curvatures 2 Dimension 2 Arbitrary dimension and codimension Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves The Menger curvature Definition (Menger 1930) The Menger curvature of three points x , y and z in R n is given by the formula 4 H 2 ( � ( x , y , z )) 1 c ( x , y , z ) := R ( x , y , z ) = | x − y || y − z || z − x | , where R ( x , y , z ) is the radius of a smallest circle passing through the points x , y and z and � ( x , y , z ) denotes the convex hull of the set { x , y , z } . Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves The Menger curvature Definition (Menger 1930) The Menger curvature of three points x , y and z in R n is given by the formula 4 H 2 ( � ( x , y , z )) 1 c ( x , y , z ) := R ( x , y , z ) = | x − y || y − z || z − x | , where R ( x , y , z ) is the radius of a smallest circle passing through the points x , y and z and � ( x , y , z ) denotes the convex hull of the set { x , y , z } . Note: Since it is defined in terms of distances and measures, it may be studied on very general metric, measure spaces! Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves Motivation More recently the Menger curvature turned out to be a useful tool (see Banavar et al. 2003 and Sutton, Balluffi, 1997) for modeling long, entangled objects like DNA molecules, protein structures or polymer chains. “The goal is to find analytically tractable notion of thickness for curves that does not rely on additional smoothness assumptions.” [Strzelecki et al. 2010] Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves Thickness for curves Let γ : S 1 → R 3 be a continuous, rectifiable curve and let Γ : S L = R / L Z → R 3 be its arclength parameterization. Definition (Gonzalez and Maddocks, 1999) The thickness of a curve γ is defined by ∆[ γ ] := inf { R (Γ( s ) , Γ( t ) , Γ( σ )) : s , t , σ ∈ S L } . Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves Thickness for curves Let γ : S 1 → R 3 be a continuous, rectifiable curve and let Γ : S L = R / L Z → R 3 be its arclength parameterization. Definition (Gonzalez and Maddocks, 1999) The thickness of a curve γ is defined by ∆[ γ ] := inf { R (Γ( s ) , Γ( t ) , Γ( σ )) : s , t , σ ∈ S L } . Theorem (Gonzalez et al. 2003) ∆[ γ ] is positive if and only if the arclength parameterization is injective of class C 1 , 1 ≃ W 2 , ∞ . Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves Thickness in variational problems Theorem (Gonzalez et al. 2002) The minimization problem � γ ∈ W 1 , q , q ∈ ( 1 , ∞ ) , I = [ a , b ] , | γ ′ | → Min ! , I with the constraints γ ( a ) = γ ( b ) , ∆[ γ ] > θ , γ ( I ) isotopic to some fixed reference curve ˜ γ ( I ) , has a solution γ ∗ and the arclength parameterization Γ ∗ ∈ C 1 , 1 . Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves Thickness in variational problems Theorem (Gonzalez et al. 2002) The minimization problem � γ ∈ W 1 , q , q ∈ ( 1 , ∞ ) , I = [ a , b ] , | γ ′ | → Min ! , I with the constraints γ ( a ) = γ ( b ) , ∆[ γ ] > θ , γ ( I ) isotopic to some fixed reference curve ˜ γ ( I ) , has a solution γ ∗ and the arclength parameterization Γ ∗ ∈ C 1 , 1 . This proves the existence of so called ideal knots , which minimize the ratio of the length to the thickness! Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves Outline One dimensional curvatures 1 Curve thickness Integral curvatures for curves Higher dimensional Menger type curvatures 2 Dimension 2 Arbitrary dimension and codimension Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves “Soft” curve energies Strzelecki, Szuma´ nska and von der Mosel suggested a different approach. The authors studied “soft” knot energies in the form of an integral of Menger curvature in some power. Definition � � � ds dt d σ M p ( γ ) := R (Γ( s ) , Γ( t ) , Γ( σ )) p , S L S L S L � � ds dt S p ( γ ) := inf σ ∈ S L R (Γ( s ) , Γ( t ) , Γ( σ )) p , S L S L � ds U p ( γ ) := inf t ,σ ∈ S L R (Γ( s ) , Γ( t ) , Γ( σ )) p . S L Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves Morrey-Sobolev imbeddings Theorem (Strzelecki, Szuma´ nska and von der Mosel, 2008) If the curve γ satisfies � � ds dt S p ( γ ) = inf σ ∈ S L R (Γ( s ) , Γ( t ) , Γ( σ )) p < ∞ S L S L for some p ∈ ( 2 , ∞ ] then the arclength parameterization Γ is injective and of class C 1 , 1 − 2 p . Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves Morrey-Sobolev imbeddings Theorem (Strzelecki, Szuma´ nska and von der Mosel, 2008) If the curve γ satisfies � � ds dt S p ( γ ) = inf σ ∈ S L R (Γ( s ) , Γ( t ) , Γ( σ )) p < ∞ S L S L for some p ∈ ( 2 , ∞ ] then the arclength parameterization Γ is injective and of class C 1 , 1 − 2 p . An analogue of the following Morrey-Sobolev imbedding W 2 , p ( R 2 ) ⊂ C 1 , 1 − 2 p ( R 2 ) , where p > 2 . Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves Morrey-Sobolev imbeddings Theorem (Strzelecki et al. 2008) If the curve γ satisfies ds dt d σ � � � M p ( γ ) = R (Γ( s ) , Γ( t ) , Γ( σ )) p < ∞ S L S L S L for some p ∈ ( 3 , ∞ ] and the arclength parameterization Γ is a local homeomorphism, then Γ ∈ C 1 , 1 − 3 p and the image Γ( S L ) is diffeomorphic to the circle S 1 . Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves Morrey-Sobolev imbeddings Theorem (Strzelecki et al. 2008) If the curve γ satisfies ds dt d σ � � � M p ( γ ) = R (Γ( s ) , Γ( t ) , Γ( σ )) p < ∞ S L S L S L for some p ∈ ( 3 , ∞ ] and the arclength parameterization Γ is a local homeomorphism, then Γ ∈ C 1 , 1 − 3 p and the image Γ( S L ) is diffeomorphic to the circle S 1 . An analogue of W 2 , p ( R 3 ) ⊂ C 1 , 1 − 3 p ( R 3 ) , whenever p > 3 . Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves Application in variational problems Let L > 0 and let k be some fixed closed curve. We set � � length ( γ ) = L γ ∈ C 0 ( S 1 , R 3 ) : C L , k := . and γ is isotopic to k Sławomir Kolasi´ nski Higher dimensional Menger curvature
One dimensional curvatures Curve thickness Higher dimensional Menger type curvatures Integral curvatures for curves Application in variational problems Let L > 0 and let k be some fixed closed curve. We set � � length ( γ ) = L γ ∈ C 0 ( S 1 , R 3 ) : C L , k := . and γ is isotopic to k Theorem (Strzelecki, Szuma´ nska and von der Mosel, 2007) Let p > 2 . In any given isotopy class represented by a closed curve k there is an arclength parameterized curve Γ ∈ C 1 , ( p − 2 ) / ( p + 4 ) ( S L , R 3 ) ∩ C L , k such that � � ds dt S p (Γ) = inf σ ∈ S L R (Γ( s ) , Γ( t ) , Γ( σ )) p = inf γ ∈ C L , k S p ( γ ) . S L S L Sławomir Kolasi´ nski Higher dimensional Menger curvature
Recommend
More recommend
