Triangulating surfaces with bounded energy

TL;DR

This paper demonstrates that surfaces with bounded Kolasinski--Menger energy can be triangulated with a controlled number of triangles, each with bounded distortion, linking geometric energy to triangulation complexity.

Contribution

It establishes a new connection between bounded energy conditions and surface triangulation with bounded distortion in Riemannian manifolds.

Findings

Triangulation number is bounded by energy and area.

Triangles are images of planar subsets with bounded distortion.

Surface energy controls triangulation complexity.

Abstract

We show that if a closed $C^1$-smooth surface in a Riemannian manifold has bounded Kolasinski--Menger energy, then it can be triangulated with triangles whose number is bounded by the energy and the area. Each of the triangles is an image of a subset of a plane under a diffeomorphism whose distortion is bounded by $\sqrt{2}$.