Immersions of compact metric spaces into Euclidean spaces

Bilipschitz Embeddings of Metric Spaces into Euclidean Spaces

When does a metric space admit a bilipschitz embedding into some finite-dimensional Euclidean space? There does not seem to be a simple answer to this question. Results of Assouad [A1], [A2], [A3] do provide a simple answer if one permits some small (“snowflake”) deformations of the metric, but unfortunately these deformations immediately disrupt some basic aspects of geometry and analysis, lik...

Minimal Immersions of Kähler Manifolds into Euclidean Spaces

We prove that a minimal isometric immersion of a Kähler-Einstein or homogeneous Kähler manifold into an Euclidean space must be totally geodesic. As an application we show that an open subset of the real hyperbolic plane RH2 cannot be minimally immersed into the Euclidean space. As another application we prove that if an irreducible Kähler manifold is minimally immersed in an Euclidean space th...

Compact Quantum Metric Spaces

We give a brief survey of many of the high-lights of our present understanding of the young subject of quantum metric spaces, and of quantum Gromov-Hausdorff distance between them. We include examples. My interest in developing the theory of compact quantum metric spaces was stimulated by certain statements in the high-energy physics and string-theory literature, concerning non-commutative spac...

Minimal Immersions of Symmetric Spaces into Spheres

Covering compact metric spaces greedily

Abstract. A general greedy approach to construct coverings of compact metric spaces by metric balls is given and analyzed. The analysis is a continuous version of Chvátal’s analysis of the greedy algorithm for the weighted set cover problem. The approach is demonstrated in an exemplary manner to construct efficient coverings of the n-dimensional sphere and n-dimensional Euclidean space to give ...