A survey of closed self-shrinkers with symmetry

TL;DR

This survey reviews known results on closed self-shrinkers in mean curvature flow, discusses symmetry-based constructions, and proposes new existence problems supported by numerical evidence for novel examples.

Contribution

It provides a comprehensive overview of closed self-shrinkers with symmetry and introduces new existence and uniqueness questions with preliminary numerical support.

Findings

Numerical evidence suggests new closed self-shrinkers with bi-rotational symmetry.

Existing techniques effectively construct self-shrinkers with classical rotational symmetry.

Open problems are proposed for future research on symmetric self-shrinkers.

Abstract

In this paper, we survey known results on closed self-shrinkers for mean curvature flow and discuss techniques used in recent constructions of closed self-shrinkers with classical rotational symmetry. We also propose new existence and uniqueness problems for closed self-shrinkers with bi-rotational symmetry and provide numerical evidence for the existence of new examples.