The Finsler Metric Obtained as the $\Gamma$-limit of a Generalised Manhattan Metric

TL;DR

This paper analytically computes the $ ext{Gamma}$-limit of length functionals for a family of two-phase Riemannian manifolds, revealing a class of piecewise affine Finsler metrics with complex discontinuities.

Contribution

It introduces a novel analytical approach to determine the $ ext{Gamma}$-limit of length functionals for two-phase Riemannian metrics, resulting in explicit Finsler metrics with intricate discontinuity structures.

Findings

Derived explicit $ ext{Gamma}$-limits for two-phase Riemannian metrics.

Identified piecewise affine Finsler metrics with infinitely many discontinuities.

Provided insights into metric behavior under microscopic geometric variations.

Abstract

The $\Gamma$-limit for a sequence of length functionals associated with a one parameter family of Riemannian manifolds is computed analytically. The Riemannian manifold is of `two-phase' type, that is, the metric coefficient takes values in $\{1,\beta\}$, with $\beta$ sufficiently large. The metric coefficient takes the value $\beta$ on squares, the size of which are controlled by a single parameter. We find a family of examples of limiting Finsler metrics that are piecewise affine with infinitely many lines of discontinuity. Such an example provides insight into how the limit metric behaves under variations of the underlying microscopic Riemannian geometry, with implications for attempts to compute such metrics numerically.