Abstract
In this paper, we first establish a constant rank theorem for the second fundamental form of the convex level sets of harmonic functions in space forms. Applying the deformation process, we prove that the level sets of the harmonic functions on convex rings in space forms are strictly convex. Moreover, we give a lower bound for the Gaussian curvature of the convex level sets of harmonic functions in terms of the Gaussian curvature of the boundary and the norm of the gradient on the boundary.
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Acknowledgements
The authors would like to thank Prof. Jiaping Wang for his help in geometry arguments, the first-named author would like to thank the encouragement of Prof. Pengfei Guan.
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Communicated by Jiaping Wang.
Research of the first author was supported by NSFC No. 11125105, by PCSIRT and “Wu Wen-Tsun Key Laboratory of Mathematics USTC Chinese Academy of Science”.
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Ma, XN., Zhang, Y. The Convexity and the Gaussian Curvature Estimates for the Level Sets of Harmonic Functions on Convex Rings in Space Forms. J Geom Anal 24, 337–374 (2014). https://doi.org/10.1007/s12220-012-9339-8
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DOI: https://doi.org/10.1007/s12220-012-9339-8