Surface Minkowski tensors to characterize shapes on curved surfaces

TL;DR

This work develops surface Minkowski tensors (SMTs) for shapes embedded on curved surfaces, introducing a defect-corrected transport to compensate for angular defects and defining irreducible SMTs whose eigenvalues $\mu_p$ quantify $p$-fold rotational symmetry on manifolds. The formalism includes a geometric setup, intrinsic functionals, and tensor definitions, with rigorous properties such as point-independence and invariance under isometries, plus explicit treatment of geodesic and regular polygons. The authors establish convergence properties for discretized representations (geodesic and straight-line polygons, triangulated surfaces) and validate the approach through numerical experiments on spheres, ellipsoids, torii, and triangulated surfaces, culminating in a biology-inspired application to cells on curved monolayers. The methodology provides a robust, scalable framework for characterizing shape anisotropy on curved spaces and enables multi-$p$ analysis to capture complex rotational symmetries in biological tissues and beyond.

Abstract

We introduce surface Minkowski tensors to characterize rotational symmetries of shapes embedded in curved surfaces. The definition is based on a modified vector transport of the shapes boundary co-normal into a reference point which accounts for the angular defect that a classical parallel transport would introduce. This modified transport can be easily implemented for general surfaces and differently defined embedded shapes, and the associated irreducible surface Minkowski tensors give rise to the classification of shapes by their normalized eigenvalues, which are introduced as shape measures following the flat-space analog. We analyze different approximations of the embedded shapes, their influence on the surface Minkowski tensors, and the stability to perturbations of the shape and the surface. The work concludes with a series of numerical experiments showing the applicability of the approach on various surfaces and shape representations and an application in biology in which the characterization of cells in a curved monolayer of cells is considered.

Figures (13)

• Figure 1: Examples of curved cell monolayers in developmental biology. (left) Lateral view of wildtype Drosophila germband extension, with permission from munster2019attachment (Figure 4). (right) Confocal image of Arabidopsis flower bud, with permission from PRUNET2016114 (Figure 1). In both images cell boundaries are shown in gray.
• Figure 2: Illustration of the general naming of embedded shapes on surfaces.
• Figure 3: Illustration of the naming: Geodesic triangle with tangents $\dot{\gamma}_i$ and angles $\alpha_i$ in the corners.
• Figure 4: Illustration of the naming: $\dot \gamma_{p;i}(0)$ in light orange and $\dot \gamma_{p;i}(l_i)$ in dark orange.
• Figure 5: Illustration of the naming: visualization of ${\bm{x}}_i-{\bm{x}}_{i+1}$ in green.

Theorems & Definitions (37)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Lemma 3.1: Point independence of the eigenvalues
• proof
• Lemma 3.2: Point independence of the eigenvectors
• proof
• Remark 3.3
• Lemma 3.4