Supercritical dual k-Minkowski flows with general prescribed data: a priori estimates and asymptotic convergence
Abstract
Let k be an integer from 1 to n, let f be a positive smooth function on the unit sphere, and consider the normalized flow X_t = -f({nu})|X|^{alpha} sigma_k({kappa}) {nu} + beta X, beta = C_n^k. In the supercritical range alpha > k+1, we prove the flow exists for all time and converges exponentially in C^{infty}, from every smooth strictly convex initial hypersurface enclosing the origin, to the unique smooth strictly convex solution of f(x)r^{alpha} sigma_k({kappa}) = beta u. The convergence is driven by the relative residual p = (log u)_t = beta - f(x)r^{alpha} {sigma}_k({kappa})/u, whose L^{infty} norm is nonincreasing and decays with the explicit exponent beta(alpha-k-1); the residual estimates are proved before, and independently of, the curvature estimates. Existence of a strictly convex solution of the stationary equation in this range, for arbitrary positive angular data, was previously obtained by Bryan-Ivaki-Scheuer through an expanding-type flow started from a barrier; the contribution here is the convergence of the normalized contracting flow from arbitrary initial data, with an explicit exponential rate, together with uniqueness of the limit.
Related
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BibTeX
@misc{sheng2026supercritical,
title = {{Supercritical dual k-Minkowski flows with general prescribed data: a priori estimates and asymptotic convergence}},
author = {Hongyi Sheng and Weimin Sheng and Jiazhuo Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35241},
eprint = {2609.35241},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35241},
}
RIS
TY - UNPB
TI - Supercritical dual k-Minkowski flows with general prescribed data: a priori estimates and asymptotic convergence
AU - Hongyi Sheng
AU - Weimin Sheng
AU - Jiazhuo Yang
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35241
UR - https://arxiv.org/abs/2609.35241
AB - Let k be an integer from 1 to n, let f be a positive smooth function on the unit sphere, and consider the normalized flow X_t = -f({nu})|X|^{alpha} sigma_k({kappa}) {nu} + beta X, beta = C_n^k. In the supercritical range alpha > k+1, we prove the flow exists for all time and converges exponentially in C^{infty}, from every smooth strictly convex initial hypersurface enclosing the origin, to the unique smooth strictly convex solution of f(x)r^{alpha} sigma_k({kappa}) = beta u. The convergence is driven by the relative residual p = (log u)_t = beta - f(x)r^{alpha} {sigma}_k({kappa})/u, whose L^{infty} norm is nonincreasing and decays with the explicit exponent beta(alpha-k-1); the residual estimates are proved before, and independently of, the curvature estimates. Existence of a strictly convex solution of the stationary equation in this range, for arbitrary positive angular data, was previously obtained by Bryan-Ivaki-Scheuer through an expanding-type flow started from a barrier; the contribution here is the convergence of the normalized contracting flow from arbitrary initial data, with an explicit exponential rate, together with uniqueness of the limit.
ER -