# Project Details

### Description

Metric graphs are considered as geometric background structures in their own right via an extension of the
combinatorial Laplacian to the Friedrich extended Laplacian on the graphs which are considered as essentially one-dimesional submanifolds in the ambient space. The vertex minimality of these graphs
guarantees not only selfadjointness of the Laplacian but also a direct comparison between functions in the ambient space and their restrictions to the graphs. This 'restriction comparison' is exåploited in this project. Metric graphs may serve as good (Haussdorff-close) approximations to surfaces in 3-space. It is conjectured that minimal metric graphs (with straight line edges) in this sense can be used to approximate minimal surfaces modulo any given $\varepsilon > 0$.
Status Finished 01/01/2002 → 01/01/2020

### Keywords

• Locally finite countable graphs
• Laplacian comparison geometry
• Minimal immersions