# Geometric Analysis and Topology Seminar

#### Morse Theory on Manifolds with Boundary and Convexity

Speaker: Gabriel Katz, William Paterson University

Location: Warren Weaver Hall 813

Date: Friday, April 6, 2007, 11 a.m.

Synopsis:

We use the gradient fields $$v$$ of nonsingular functions $$f$$ on compact 3-folds $$X$$ with boundary to generate their spines $$K(f,v)$$. We study the transformations of $$K(f,v)$$ that are induced by deformations of the data $$(f,v)$$. Also, we link the Matveev complexity $$c(X)$$ of $$X$$ with counting the double-tangent trajectories of the $$v$$-flow. These are tangent to the boundary $$\partial X$$ at a pair of distinct points. Let $$MC(X)$$ be the minimum number of such trajectories, minimum being taken over all nonsingular $$v$$'s. We call $$MC(X)$$ the Morse complexity of $$X$$. Next, we prove that there are only finitely many irreducible and boundary irreducible $$X$$ with no essential annuli of bounded Morse complexity. In particular, there exists only finitely many hyperbolic manifolds $$X$$ with bounded $$MC(X)$$. For such $$X$$, their normalized hyperbolic volume gives a lower bound of $$MC(X)$$. Also, if an irreducible and boundary irreducible $$X$$ with no essential annuli admits a nonsingular gradient flow with no double-tangent trajectories, then $$X$$ is a standard ball.

All these results rely on a careful study of the stratified geometry of $$\partial X$$ relative to the $$v$$-flow. It is characterized by failure of $$\partial X$$ to be convex with respect to a generic flow $$v$$. It turns out, that convexity or its lack have profound influence on the topology of $$X$$. This phenomenon is in the focus of the talk.