Let a closed ‐dimensional manifold, be a closed manifold, and let for . We extend the monumental work of Sacks and Uhlenbeck by proving that if , then there exists a minimizing ‐harmonic map homotopic to . If , then we prove that there exists a ‐harmonic map from to in a generating set of . Since several techniques, especially Pohozaev‐type arguments, are unknown in the fractional framework (in particular, when , one cannot argue via an extension method), we develop crucial new tools that are interesting on their own: such as a removability result for point singularities and a balanced energy estimate for nonscaling invariant energies. Moreover, we prove the regularity theory for minimizing ‐maps into manifolds.
Functional Maps Representation On Product Manifolds
We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.
more » « less- Award ID(s):
- 1838071
- NSF-PAR ID:
- 10461152
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Computer Graphics Forum
- Volume:
- 38
- Issue:
- 1
- ISSN:
- 0167-7055
- Page Range / eLocation ID:
- p. 678-689
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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