On the Existence of a Closed, Embedded, Rotational $\lambda$-Hypersurface

In this paper we show the existence of a closed, embedded $\lambda$-hypersurfaces $\mathrm{\Sigma }\subset {\mathbb{R}}^{2n}$. The hypersurface is diffeomorhic to ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$ and exhibits $SO(n)\times SO(n)$ symmetry. Our approach uses a “shooting method” similar to the approach used by McGrath in constructing a generalized self-shrinking torus solution to mean curvature flow. The result generalizes the $\lambda$-torus found by Cheng and Wei.