Evolution of hypersurfaces in (n+1)-dimensional light-cone

2026/07/29 by Xinjie Jiang, Shengliang Pan, Yun Yang
Mathematics · #math.DG

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arxiv created 2026/07/29 · arxiv updated 2026/07/30

Abstract

In this paper, we investigate the evolutionary processes of hypersurfaces within half of the (n+1)-dimensional light-cone. Depending on the evolutionary processes, our focus extends to exploring variational problems associated with a smooth function f(S₁,⋯,S_n), where each S_r denotes the r-th elementary symmetric polynomial, defined as the sum of all possible products of r distinct principal curvatures. We present several fundamental properties related to these variational problems. Furthermore, we examine a curvature-type flow defined locally within the light-cone, establishing its perpetual existence and smooth convergence to a circle whose length is preserved and equal to that of the initial curve.

Citations

• Ricci flow on surfaces along the standard lightcone in the 3+1-Minkowski spacetime
• Minimal hypersurfaces and geometric inequalities
• The isoperimetric inequality for a minimal submanifold in Euclidean space
• A volume preserving flow and the isoperimetric problem in warped product spaces
• A mean curvature type flow in space forms
• Embedded minimal tori in S³ and the Lawson conjecture
• Embedded minimal tori in S3 and the Lawson conjecture
• Min-Max theory and the Willmore conjecture
• The Min-Max theory and the Willmore conjecture
• A sharp bound for the area of minimal surfaces in the unit ball
• Volume preserving mean curvature flow in the Hyperbolic space
• Curvature estimates for minimal hypersurfaces

Related

• A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone
• Evolution of starshaped hypersurfaces by general curvature functions
• Evolution of non-compact hypersurfaces by inverse mean curvature
• The mean curvature flow of submanifolds of high codimension
• Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone