Ferrand invariant

In mathematics , the Ferrand invariant is an invariant named after Jacqueline Ferrand , which is particularly useful when studying conformal mappings .

definition

The Ferrand invariant of four points in a -dimensional Riemannian manifold is the infimum of the capacity over all disjoint , compact sets with and . ${\displaystyle j(x,y,z,t)}$ ${\displaystyle x,y,z,t}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle c(F_{0},F_{1})}$ ${\displaystyle x,y\in F_{0}}$ ${\displaystyle z,t\in F_{1}}$

The capacity of two disjoint sets is defined as the infimum of over all smooth functions with . ${\displaystyle \int \vert du\vert ^{n}}$ ${\displaystyle u|_{F_{0}}\equiv 0,u|_{F_{1}}\equiv 1}$

The invariant is infinite, if or with or matches. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle t}$

properties

• The invariant is always positive: . ${\displaystyle j(x,y,z,t)\not =0}$
• By on the complement of defined metric is on the Riemannian manifold defines the conventional topology. ${\displaystyle \left\{z,t\right\}}$ ${\displaystyle d(x,y):=j(x,y,z,t)^{1/(1-n)}}$ ${\displaystyle M\setminus \left\{z,t\right\}}$
• The metric tends towards infinity when it strives towards (for fixed ) towards or (for fixed ) towards . ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle z}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle t}$

Applications

• The Ferrand invariant allows estimations for the continuity modulus of conforming maps .
• With the help of the invariant, Ferrand proved that the group of conformal mappings fixing three points y, z, t is a compact group .
• For a divergent sequence of conformal mappings one can show that there are at most two possible limit values ​​of sequences . If there are actually two such limit values , then a subsequence of sends the complement of each neighborhood from to arbitrarily small neighborhoods from . Diverging sequences of conforming images always show a north-south dynamic . ${\displaystyle f_{i}}$ ${\displaystyle f_{i}(x_{i})}$ ${\displaystyle z\not =t}$ ${\displaystyle f_{i}}$ ${\displaystyle z}$ ${\displaystyle t}$
• From the previous point it follows that manifolds with diverging sequences of conformal mappings must simply be connected and conformally flat, i.e. they must in particular be a sphere . With this argument, Ferrand proved the Lichnerowicz conjecture : a compact Riemannian manifold with a non-compact set of conformal transformations must be conformally equivalent to the unit sphere.
• Estimates for the Ferrand invariant are an important tool in regularity theory of quasi-conformal maps . In particular, it is used for the proof of the quasi-isometric rigidity of hyperbolic buildings and the characterization of 2-dimensional spheres down to quasi-symmetry .

literature

• J. Ferrand: Transformations conformes et quasi-conformes des variétés riemanniennes compactes (demonstration de la conjecture de A. Lichnerowicz). Acad. Roy. Belg. Cl. Sci. Mém. Coll. in – 8deg (2) 39, no. 5, 44 pp. (1971).

Web links

• Pierre Pansu : Jacqueline Ferrand and her oeuvre (Notices of the AMS)

Individual evidence

1. ↑ M. Bourdon , H. Pajot : Quasi-conformal rigidity and hyperbolic geometry , in Rigidity in dynamics and geometry , Springer 2002
2. ↑ M. Bonk , B. Kleiner : Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math. 150, 127-183 (2002)