# Variational and rigidity properties of the Hawking mass

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

This Ph.D. project is in pure mathematics, with motivations coming from mathematical physics (in particular mathematical general relativity), and is aimed to prove variational and rigidity properties of the Hawking mass. The Hawking mass is a fundamental concept in (mathematical) general relativity modeling the quasi-local mass contained inside a solid domain in a riemannian space-like slice of a space-time.

The hawking mass has been a key technical tool in modern mathematical general relativity, for instance the proof of Huisken-Ilmanen of the celebrated Penrose inequality (riemannian case) is based in the monotonicity of the Hawking mass under inverse mean curvature flow.

The project will lie at the triple point between differential geometry, partial differential equations, and mathematical general relativity. In particular, non linear analysis and partial differential equations will play a key technical role to attack the problems which are motivated mainly from differential geometry and mathematical physics.

The first goal of the project is to show that if for every point in a riemannian space-like slice there exists a small neighborhood where the supremum of the Hawking mass is zero, then all the space must be flat. This can be seen as a local-to-global property of the Hawking mass.

In order to solve such a problem the student will analyze the behavior of the Hawking mass at small scales, linking the Hawking mass of a small surface with the geometry of the ambient space.

More broadly the goal of the thesis is to explore the variational features of the Hawking mass, investigating local maximizers and more generally (constrained) critical points of such a functional under natural geometric conditions (like curvature restrictions).

The hawking mass has been a key technical tool in modern mathematical general relativity, for instance the proof of Huisken-Ilmanen of the celebrated Penrose inequality (riemannian case) is based in the monotonicity of the Hawking mass under inverse mean curvature flow.

The project will lie at the triple point between differential geometry, partial differential equations, and mathematical general relativity. In particular, non linear analysis and partial differential equations will play a key technical role to attack the problems which are motivated mainly from differential geometry and mathematical physics.

The first goal of the project is to show that if for every point in a riemannian space-like slice there exists a small neighborhood where the supremum of the Hawking mass is zero, then all the space must be flat. This can be seen as a local-to-global property of the Hawking mass.

In order to solve such a problem the student will analyze the behavior of the Hawking mass at small scales, linking the Hawking mass of a small surface with the geometry of the ambient space.

More broadly the goal of the thesis is to explore the variational features of the Hawking mass, investigating local maximizers and more generally (constrained) critical points of such a functional under natural geometric conditions (like curvature restrictions).

## People |
## ORCID iD |

Andrea Mondino (Primary Supervisor) | |

Aidan Browne (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/N509796/1 | 01/10/2016 | 30/09/2021 | |||

1935375 | Studentship | EP/N509796/1 | 02/10/2017 | 30/09/2021 | Aidan Browne |