### Abstract

A given metric (length-) space $X$ (whether compact or not) is roughly isometric to any one of its Kanai graphs $G$, which in turn can be {\em{geometrized}} by considering each edge of $G$ as a 1-dimensional manifold with an associated metric $g$ giving the 'correct' length of the edge.
In this talk we will mainly be concerned with {\em{minimal}} isometric immersions of such geometrized approximations $(G, g)$ of $X$ into Riemannian manifolds $N$ with bounded curvature.
When such an immersion exists, we will call it an $X$-web in $N$.
Such webs admit a natural 'geometric' extension of the intrinsic combinatorial discrete Laplacian, and we will show that they share several analytic and geometric properties with their smooth (minimal submanifold) counterparts in $N$. The intrinsic properties thus obtained may hence serve as roughly invariant descriptors for the original metric space $X$.

Original language | English |
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Publication date | 2008 |

Publication status | Published - 2008 |

Event | Workshop on Distance Geometry - Salzburg, Austria Duration: 5 May 2008 → 8 May 2008 |

### Workshop

Workshop | Workshop on Distance Geometry |
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Country | Austria |

City | Salzburg |

Period | 05/05/2008 → 08/05/2008 |

### Keywords

- Distance geometry

## Cite this

Markvorsen, S. (2008).

*Roughly isometric minimal immersions into Riemannian manifolds*. Abstract from Workshop on Distance Geometry, Salzburg, Austria.