Search

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson [7] as a generalization of Thurston's circle packing metric [34] . They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo [22] proved the infinitesimal rigidity and then Luo [27] proved the global rigidity. In this paper, based on an observation of Zhou [37] , we prove this conjecture for inversive distance in ( − 1 , + ∞ ) by variational principles. We also study the global rigidity of a combinatorial curvature introduced in [14,16,19] with respect to the inversive distance circle packing metrics where the inversive distance is in ( − 1 , + ∞ ) . [ABSTRACT FROM AUTHOR]

Solving a well-known problem of Maharam, Talagrand [18] constructed an exhaustive non uniformly exhaustive submeasure, thus also providing the first example of a Maharam algebra that is not a measure algebra. To each exhaustive submeasure one can canonically assign a certain countable ordinal, its exhaustivity rank. In this paper, we use carefully constructed Schreier families and norms derived from them to provide examples of exhaustive submeasures of arbitrary high exhaustivity rank. This gives rise to uncountably many non isomorphic separable atomless Maharam algebras. [ABSTRACT FROM AUTHOR]

This is the second of two papers, in which we study the problem of prescribing Webster scalar curvature on the CR sphere as a given function f . Using the Webster scalar curvature flow, we prove an existence result under suitable assumptions on the Morse indices of f . [ABSTRACT FROM AUTHOR]