\(\text{CAT}(\kappa)\)-spaces: construction and concentration
This dense article surveys many past and current work, many (but not all) by the author, on a circle of ideas going from the geometry of singular spaces with curvature bounded from above to geometric group theory, concentration results, spectral estimates, diffusion and harmonic maps, construction of negatively (or non-positively) curved spaces through geometric constructions, etc. Although many of the proofs are only sketched, the paper contains a wealth of beautiful ideas and gives many hints of the relations between them and between different themes in mathematics. Rather than trying to give a faithful description of the content, which appears to be especially difficult here, it might be useful for the potential reader to cite the titles of some of the paragraphs, which indicate well the mathematical themes that this paper touches. \S 1. Metrics and geodesics. \S 3. Model-spaces. \S 4. Comparison relations between the model spaces. \S 5. Positivity relations. \S 6. Wirtinger inequalities. \S 8. Geometric application. \S 13. Concentration in smooth \(X\). \S 14. Diffusion, codiffusion and harmonic maps. \S 15. Geodesic triangles and CAT(\(\kappa\))-spaces. \S 16. Geodesic convexity and convex gluing of spaces. \S 17. Convexity and CAT(\(\kappa\))-convexity. \S 20. Ramified convers. \S 21. On construction of polyhedra \(X\) with \(K\leq 0\) for dim\((X)\geq 3\). \S 22. Assembling \((K\leq 0)\)-spaces over geodesic graphs. \S 24. Effective universal coverings of spaces with \(K\leq 0\). \S 25. Filling closed curves by disks in CAT-spaces. \S 26. CAT-families of groups. \S 29. Coning CAT-spaces and their subspaces.
scientific article; zbMATH DE number 2182762