Schoenflies Spheres as Boundaries of Bounded Unstable Manifolds in Gradient Sturm Systems

Abstract

Let $$v$$ be a hyperbolic equilibrium of a smooth finite-dimensional gradient or gradient-like dynamical system. Assume that the unstable manifold $$W$$ of $$v$$ is bounded, with topological boundary $$\Sigma = \partial \!W:= (clos W)\backslash W$$ . Then $$\Sigma $$ need not be homeomorphic to a sphere, or to any compact manifold. However, consider PDEs $$\begin{aligned} u_{t} = u_{xx} + f(x,u,u_x) \end{aligned}$$ of Sturm type, i.e. scalar reaction–advection–diffusion equations in one space dimension. Under separated boundary conditions on a bounded interval this defines a gradient dynamical system. For such gradient Sturm systems, we show that the eigenprojection P $$\Sigma $$ of $$\Sigma $$ onto the unstable eigenspace of $$v$$ is homeomorphic to a sphere. In particular this excludes complications like lens spaces and Reidemeister torsion. Excluding Schoenflies complications like Alexander horned spheres, we also show that both the interior domain $$PW$$ of P $$\Sigma $$ and the one-point compactified exterior domain in the tangential eigenspace are homeomorphic to open balls. Our results are based on Sturm nodal properties.