# 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.