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

Xinjie Jiang; Shengliang Pan; Yun Yang

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

Differential Geometry 2026-07-29 v1

Authors: Xinjie Jiang , Shengliang Pan , Yun Yang

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 $f(S_1,\cdots,S_n)$ , 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.

Keywords

hypersurface evolution light-cone curvature flow variational problems elementary symmetric polynomials

Cite

@article{arxiv.2607.26462,
  title  = {Evolution of hypersurfaces in $(n+1)$-dimensional light-cone},
  author = {Xinjie Jiang and Shengliang Pan and Yun Yang},
  journal= {arXiv preprint arXiv:2607.26462},
  year   = {2026}
}