# The geometry of manifolds with asymptotically flat ends

**Time: **
Fri 2023-09-15 13.15

**Location: **
3418, Lindstedtsvägen 25, Stockholm

**Language: **
English

**Subject area: **
Mathematics

**Doctoral student: **
Bernardo Hipólito Fernandes
, Matematik (Avd.)

**Opponent: **
Stephen McCormick, Matematik (Inst.)

**Supervisor: **
Mattias Dahl, Matematik, Matematik (Avd.)

QC 2023-08-21

## Abstract

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.