Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Novel Boundary Conditions for the Ricci Flow

Rasmus Jouttijärvi

Abstract

If we want to deform a compact Riemannian manifold with boundary using Ricci flow, we first need to decide on appropriate boundary conditions. We would like these conditions to reflect the geometric nature of the flow and allow for a variety of initial data. Importantly, the conditions should be compatible with the expected evolution of Einstein metrics. We propose it is natural to choose those conditions, for which the first variation of certain functionals, such as the Einstein-Hilbert action and Perelmans lambda-functional, does not admit a boundary term. We provide a proof of the short term existence of solutions of the initial boundary value problem, under these conditions.

Novel Boundary Conditions for the Ricci Flow

Abstract

If we want to deform a compact Riemannian manifold with boundary using Ricci flow, we first need to decide on appropriate boundary conditions. We would like these conditions to reflect the geometric nature of the flow and allow for a variety of initial data. Importantly, the conditions should be compatible with the expected evolution of Einstein metrics. We propose it is natural to choose those conditions, for which the first variation of certain functionals, such as the Einstein-Hilbert action and Perelmans lambda-functional, does not admit a boundary term. We provide a proof of the short term existence of solutions of the initial boundary value problem, under these conditions.
Paper Structure (6 sections, 17 theorems, 228 equations)

This paper contains 6 sections, 17 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $\lambda$-functional
  4. Function Spaces and Boundary Value Problems
  5. Ricci-deTurck Flow
  6. Ricci Flow

Key Result

Theorem 1.1

Let $g$ be a smooth compatible Riemannian metric, then, for $p>n$, there exist a unique solution to the initial boundary value problem Such that $\partial_t^{m}g_t\in L^p\left( M\times [0,T]\right)$ for $0\leq m\leq 2$ and $g_t\in W^{4,p}\left( M\right)$ for all $t\in [0,T]$.

Theorems & Definitions (42)

  • Theorem 1.1: Existence and uniqueness
  • Theorem 1.2: Boundary regularity
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 3.1: The $\lambda$-functional
  • Lemma 3.2
  • proof
  • Proposition 3.3: First variation of $\lambda$
  • ...and 32 more