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

TL;DR

This paper proves the existence of a new class of closed, embedded rotational λ-hypersurfaces in Euclidean space, expanding the understanding of such geometric structures with specific symmetry properties.

Contribution

It introduces a novel existence proof for a specific λ-hypersurface with SO(n)×SO(n) symmetry, generalizing previous λ-torus solutions using a shooting method approach.

Findings

Existence of a closed, embedded λ-hypersurface diffeomorphic to S^{n-1}×S^{n-1}×S^1.

Hypersurface exhibits SO(n)×SO(n) symmetry.

Generalizes the λ-torus by Cheng and Wei.

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.