Abstract
A well studied classical problem is the harmonicity of functions satisfying the restricted mean-value property (RMVP). While this has so far been studied mainly for domains in \(\mathbb {R}^n\), we consider this problem in the general setting of domains in Riemannian manifolds, and obtain results generalizing classical results of Fenton. We also obtain a result for complete, simply connected Riemannian manifolds of pinched negative curvature where there is no restriction on the radius function in the RMVP.
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Acknowledgements
The authors would like to thank Swagato K. Ray for suggesting the problems. The second author is supported by a Research Fellowship of Indian Statistical Institute.
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Biswas, K., Dewan, U. Restricted Mean Value Property on Riemannian manifolds. J Geom Anal 34, 108 (2024). https://doi.org/10.1007/s12220-024-01555-3
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DOI: https://doi.org/10.1007/s12220-024-01555-3
Keywords
- Restricted mean value property
- Harmonic functions
- Harmonic measure
- Negatively curved Hadamard manifolds