On Critical nets in $\mathbb{R}^k$
| Autor: | Gournay, Antoine, Memarian, Yashar |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s10711-020-00556-0 |
| Popis: | Critical nets in $\mathbb{R}^k$ (sometimes called geodesic nets) are embedded graph with the property that their embedding is a critical point of the total (edge) length functional and under the constraint that certain 1-valent vertices (leaves) have a fixed position. In contrast to what happens on generic manifolds, we show that, if n is the number of 1-valent vertices, the total length of the edges not incident with a 1-valent vertex is bounded by rn (where r is the outer radius), the degree of any vertex is bounded by n and that the number of edges (and hence the number of vertices) is bounded by nl where l is related to the combinatorial diameter of the graph. Comment: 19 pages, 1 figure |
| Databáze: | arXiv |
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