Abstract
Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of a harmonic function along the gradient flow is determined by the mean curvature of the level sets that the flow intersects.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Flatto, L., D.J. Newman, and H.S. Shapiro. 1966. The Level Curves of Harmonic Functions. Transactions of the American Mathematical Society 123: 425–436.
Jerrard, R.P., and L.A. Rubel. 1963. On the Curvature of the Level Lines of a Harmonic Function. Proceedings of the American Mathematical Society 14: 29–32.
Funding
There is no source of funding for this article.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that this research involves no conflict of interest.
Ethical approval
This article does not contain any studies with human participants or animals.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ding, P. The family of level sets of a harmonic function. J Anal 28, 895–904 (2020). https://doi.org/10.1007/s41478-019-00218-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41478-019-00218-9