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Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine Existence
Abstract
This paper uses probability theory to provide an approach to the Dirichlet problem for harmonic maps. The probabilistic tools used are those of manifold-valued Brownian motion and Γ-martingales. Probabilistic proofs are given of uniqueness, continuous dependence on data, and existence of generalized solutions (finely harmonic maps), for target domains which have convex geometry. (This class of domain includes all regular geodesic balls.) Links are made with new results in other fields: the Γ-martingale Dirichlet problem, Riemannian centres of mass (herein termed ‘Karcher means’), and the existence of certain convex functions. Future prospects include the development of the probabilistic approach in order to formulate the notion of a harmonic map defined on a fractal and to attack the corresponding Dirichlet problem.
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There are results on the non-emptiness and the uniqueness of Fréchet means for manifolds.
“There are results on the non-emptiness and the uniqueness of Fr´ echet means for manifolds (Karcher 1977, Kendall 1990).”
From Topological pattern recognition for point cloud data (2014) · cites this paper as Kendall 1990
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Metadata
- Authors
- Kendall, W. S.
- Affiliations
- University of Warwick (GB)
- Corresponding author
- Kendall, Wilfrid S.
- ORCID
- Kendall, Wilfrid S.
- Venue
- Proceedings of the London Mathematical Society
- Publisher
- Wiley
- Published
- 1990-09
- Volume
- s3-61
- Issue
- 2
- Pages
- 371-406
- ISSN
- 0024-6115 (ISSN-L)
- OpenAlex
- W2040125300
- Cited by
- 281
- References
- 0
- Citations by year
- 2026: 4 · 2025: 5 · 2024: 3 · 2023: 10 · 2022: 7 · 2021: 12 · 2020: 13 · 2019: 13 · 2018: 9 · 2017: 9 · 2016: 13 · 2015: 15 · 2014: 10 · 2013: 17 · 2012: 17
- FWCI
- 7.93
- Type
- journal article
- Language
- en
- Open access
- closed
- Licence
- Locations
- Proceedings of the London Mathematical Society (publishedVersion)
- HighWire Press Open Archive (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Mathematics › Applied Mathematics › Nonlinear Partial Differential Equations
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
- Keywords
- harmonic maps, Dirichlet problem, convex domains, geodesic balls, continuous dependence on data, weak solutions, Karcher mean, convex functions
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