Non-homogeneous curvature flows in a hemisphere
Abstract
Let S^{n+1}_{+} be the open hemisphere of the unit sphere S^{n+1} centred at o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^{alpha} {nu} of smooth, closed, strictly convex hypersurfaces enclosing o, where r is the geodesic distance to o. We consider both the supercritical regime beta>1+k {alpha} and the critical regime beta=1+k{alpha}, where beta is the growth order of the profile at the origin, f(r) almost equals r^{beta} as r descends to 0. Under the structural condition that f^{1/(1+k{alpha})} is convex, we prove long-time existence and preservation of strict convexity. The normalized radial function converges smoothly and exponentially to a constant: to 1 and to R_{infty}>0, resp. in different two cases. Thus the normalized radial graphs become round, while the original hypersurfaces contract to o.
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BibTeX
@misc{sheng2026nonhomogeneous,
title = {{Non-homogeneous curvature flows in a hemisphere}},
author = {Hongyi Sheng and Weimin Sheng and Jiazhuo Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31023},
eprint = {2609.31023},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31023},
}
RIS
TY - UNPB
TI - Non-homogeneous curvature flows in a hemisphere
AU - Hongyi Sheng
AU - Weimin Sheng
AU - Jiazhuo Yang
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31023
UR - https://arxiv.org/abs/2609.31023
AB - Let S^{n+1}_{+} be the open hemisphere of the unit sphere S^{n+1} centred at o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^{alpha} {nu} of smooth, closed, strictly convex hypersurfaces enclosing o, where r is the geodesic distance to o. We consider both the supercritical regime beta>1+k {alpha} and the critical regime beta=1+k{alpha}, where beta is the growth order of the profile at the origin, f(r) almost equals r^{beta} as r descends to 0. Under the structural condition that f^{1/(1+k{alpha})} is convex, we prove long-time existence and preservation of strict convexity. The normalized radial function converges smoothly and exponentially to a constant: to 1 and to R_{infty}>0, resp. in different two cases. Thus the normalized radial graphs become round, while the original hypersurfaces contract to o.
ER -