The geometry of manifolds with asymptotically flat ends
Time: Fri 2023-09-15 13.15
Location: 3418, Lindstedtsvägen 25, Stockholm
Subject area: Mathematics
Doctoral student: Bernardo Hipólito Fernandes , Matematik (Avd.)
Opponent: Stephen McCormick, Matematik (Inst.)
Supervisor: Mattias Dahl, Matematik, Matematik (Avd.)
This monograph thesis is divided into two chapters.
In the first, "A study of ALF structures", we prove that the structure at infinity of a ALF manifold is essentially unique, i.e. any structures associated to a complete non-compact Riemannian manifold that is asymptotic to a circle fibration over an Euclidean base, with fibres of asymptotically constant length, are related by a rigid motion plus low-order terms. The proof is based on showing the existence of harmonic coordinate-like functions.
In the second, "A study of fibred boundary metrics", we show the existence of a unique transverse diffeomorphism associated to a fibred boundary metric, i.e. a map that transforms a fibred boundary metric into another with no transverse components. Furthermore, we demonstrate that a diffeomorphism that maps a fixed fibred boundary metric into another can be uniquely decomposed as a composition of an isometry on the boundary and a small diffeomorphism. We present several examples of such transverse diffeomorphisms together with their decompositions.
We conclude this second chapter by introducing the notion of linear mass at infinity of a fibred boundary metrics, and give a full classification of the 3-dimensional case associated to asymptotically euclidean fibred boundary metrics.