球面中horo-凸超曲面的保持quermassintegral的曲率流
The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere
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中文总结 AI 辅助
本研究提出球面中horo-凸超曲面的完全非线性曲率流，可保持任意指定球面quermassintegral，证明了horo-凸性保持、长时间存在及指数收敛到对应测地球面。
中文摘要 AI 辅助
我们引入了球面中horo-凸超曲面的完全非线性曲率流，该流可保持任意指定的球面quermassintegral。这种非局部归一化是相对于固定的环境原点定义的，我们证明了演化过程在所有时刻都由单个光滑的固定原点方程控制。对于满足凹性和逆凹性的单调齐次曲率函数，我们证明了horo-凸性的保持性、一致曲率夹紧性，以及非局部系数的直接估计。随后，Tso型论证给出了全局曲率界和长时间存在性。我们进一步证明了无迹第二基本形式的指数衰减性，以及流以指数级C^∞收敛到以固定原点为中心的测地球面，其半径由被保持的quermassintegral决定。
英文摘要
We introduce fully nonlinear curvature flows of horo-convex hypersurfaces in the sphere that preserve an arbitrarily prescribed spherical quermassintegral. The non-local normalization is defined relative to a fixed ambient origin, and we prove that the evolution is governed by a single smooth fixed-origin equation for all time. For monotone, homogeneous curvature functions satisfying concavity and inverse concavity, we establish preservation of horo-convexity, uniform curvature pinching, a direct estimate for the non-local coefficient. A Tso-type argument then yields global curvature bounds and long-time existence. We further prove exponential decay of the traceless second fundamental form and exponential $C^\infty$ convergence to the geodesic sphere centered at the fixed origin whose radius is determined by the preserved quermassintegral.
