超临界对偶 k-Minkowski 流与一般给定数据：先验估计与渐近收敛
Supercritical dual k-Minkowski flows with general prescribed data: a priori estimates and asymptotic convergence
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- Westlake University（西湖大学）
- Zhejiang University（浙江大学）
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中文总结 AI 辅助
本文研究超临界对偶 k-Minkowski 流的长时间行为，证明其从任意严格凸初始曲面指数收敛到唯一严格凸稳态解，并给出显式收敛速率与唯一性。
中文摘要 AI 辅助
设 k 为从 1 到 n 的整数，f 为单位球面上的正光滑函数，考虑归一化流 X_t = -f({nu})|X|^{alpha} sigma_k({kappa}) {nu} + beta X，其中 beta = C_n^k。在超临界范围 alpha > k+1 内，我们证明该流对所有时间存在，并从任意包含原点的光滑严格凸初始超曲面出发，以 C^{infty} 范数指数收敛到方程 f(x)r^{alpha} sigma_k({kappa}) = beta u 的唯一光滑严格凸解。收敛由相对残差 p = (log u)_t = beta - f(x)r^{alpha} {sigma}_k({kappa})/u 驱动，其 L^{infty} 范数非增且以显式指数 beta(alpha-k-1) 衰减；残差估计在曲率估计之前且独立于曲率估计而证明。在该范围内，对任意正角数据，严格凸解的适定性此前由 Bryan-Ivaki-Scheuer 通过从障碍开始的扩张型流获得；本文的贡献在于从任意初始数据出发的归一化收缩流的收敛性，具有显式指数速率，以及极限的唯一性。
英文摘要
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.
