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

Abstract

In this paper we show the existence of a closed, embedded $\lambda$-hypersurfaces $\Sigma \subset \mathbb{R}^{2n}$. 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.