Circular $m$-functions

Abstract

Circular $m$-functions are introduced on smooth manifolds with boundary. We study the distribution of their critical circles and construct an example of a four-dimensional manifol $dM^4$ with boundary $∂M^4$ that satisfies the condition $ξ(∂M 4) = ξ(M^4,∂M^4) = 0$ but does not contain any circularm-function. We prove that a manifold with boundary $M^n (n ≥ 5)$ such that $ξ(∂M^n , ∂M^n ) = 0$ always contains a circularm-function without critical points in the interior manifold.