2020/06/30 by Robert Cardona, Cardona, Robert · 1 citation
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology
paper · doi:10.48550/arxiv.2006.16880
We construct non-vanishing steady solutions to the Euler equations (for some metric) with analytic Bernoulli function in each three-manifold where they can exist: graph manifolds. Using the theory of integrable systems, any admissible Morse-Bott function can be realized as the Bernoulli function of some non-vanishing steady Euler flow. This can be interpreted as an inverse problem to Arnold's structure theorem and yields as a corollary the topological classification of such solutions. Finally, we prove that the topological obstruction holds without the non-vanishing assumption: steady Euler flows with a Morse-Bott Bernoulli function only exist on graph three-manifolds.