Constructing Reference Metrics on Multicube Representations of Arbitrary Manifolds

TL;DR

This paper presents a general, automated method for constructing and smoothing reference metrics on multicube representations of arbitrary manifolds, validated through numerical tests on 2D manifolds with various topologies.

Contribution

It introduces a novel procedure for automatically constructing reference metrics on multicube manifolds of any topology, including smoothing via Ricci flow.

Findings

Constructed reference metrics satisfy Gauss-Bonnet identity numerically.

Ricci flow smoothing produces metrics with constant scalar curvature.

Method successfully applied to 2D manifolds with genera 0 to 5.

Abstract

Reference metrics are used to define the differential structure on multicube representations of manifolds, i.e., they provide a simple and practical way to define what it means globally for tensor fields and their derivatives to be continuous. This paper introduces a general procedure for constructing reference metrics automatically on multicube representations of manifolds with arbitrary topologies. The method is tested here by constructing reference metrics for compact, orientable two-dimensional manifolds with genera between zero and five. These metrics are shown to satisfy the Gauss-Bonnet identity numerically to the level of truncation error (which converges toward zero as the numerical resolution is increased). These reference metrics can be made smoother and more uniform by evolving them with Ricci flow. This smoothing procedure is tested on the two-dimensional reference metrics constructed here. These smoothing evolutions (using volume-normalized Ricci flow with DeTurck gauge fixing) are all shown to produce reference metrics with constant scalar curvatures (at the level of numerical truncation error).