Shapes of large, static soap bubbles

Standard predictions induced by the balance of surface tension and pressure dictate that static soap bubbles must be spherical. However, definite non-spherical shapes appear in large bubbles, where noticeable oblate or prolate deformations occur. Gravity is the principal cause of such deformations, and multiple approaches for including its influence appear in recent literature. This paper derives a general surface-theoretic model by applying asymptotic and variational methods to a fully three-dimensional set-up where the soap bubble is a finite-thickness film. The procedure illuminates implicit assumptions, clarifying the discrepancies seen in previous models. Then the model is studied in four physical situations. In three of these situations, results show that there is a maximum stable span and volume of the soap bubbles, implying that their behaviour is qualitatively more similar to liquid drops than standard soap bubbles. Also, the model presented is directly analogous to the two-dimensional version of a hanging chain, and the derived predictions give practical insights into the construction of heavy containment vessels.

Twisted signatures of flag manifolds

A product formula for some twisted signatures of flag manifolds is proved. The result is used to compute twisted signatures of some flag manifolds from those of Grassmannians, and by that to deduce some upper bounds of the stable span.