differential geometry

Evolution of hypersurfaces in -dimensional light-cone

arXiv:2607.26462

summary

The paper studies how hypersurfaces evolve inside the half of an (n+1)-dimensional light‑cone, analyzes variational problems involving elementary symmetric functions of principal curvatures, and proves that a curvature‑type flow exists for all time and smoothly converges to a length‑preserving circle.

Abstract

In this paper, we investigate the evolutionary processes of hypersurfaces within half of the -dimensional light-cone. Depending on the evolutionary processes, our focus extends to exploring variational problems associated with a smooth function , where each denotes the -th elementary symmetric polynomial, defined as the sum of all possible products of 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.

Topics & keywords

#hypersurface evolution#light-cone geometry#curvature flow#variational problems#symmetric polynomialselementary symmetric polynomialsprincipal curvaturesgeometric flowsmooth convergencelength preservation
Evolution of hypersurfaces in $(n+1)$-dimensional light-cone · wovepaper