### Abstract

Let (Formula presented.) be a supercritical Sobolev map from a domain (Formula presented.) in (Formula presented.) into a complete separable metric space. For a fixed integer (Formula presented.), (Formula presented.), and (Formula presented.), we estimate from above the Hausdorff dimension of the set of elements (Formula presented.) in the Grassmannian (Formula presented.) (equipped with a Riemannian metric) such that (Formula presented.) has positive (Formula presented.)-dimensional Hausdorff measure. The proof relies heavily on the homogeneous structure of both (Formula presented.) and the Stiefel manifold (Formula presented.) of orthogonal injections of (Formula presented.) into (Formula presented.). A novel feature of the proof is a Morrey–Sobolev-type embedding theorem on the product manifold (Formula presented.), valid for Sobolev maps which factor through the evaluation map (Formula presented.).

Original language | English (US) |
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Pages (from-to) | 505-523 |

Number of pages | 19 |

Journal | Computational Methods and Function Theory |

Volume | 14 |

Issue number | 2-3 |

DOIs | |

State | Published - Oct 31 2014 |

### Keywords

- Grassmannian manifold
- Hausdorff dimension
- Sobolev mapping
- Stiefel manifold
- potential theory

### ASJC Scopus subject areas

- Analysis
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Computational Methods and Function Theory*,

*14*(2-3), 505-523. https://doi.org/10.1007/s40315-014-0058-y