Minimal immersions of Riemannian manifolds in products of space forms

TL;DR

This paper extends classical results on minimal immersions to cylinders and tori, focusing on Euclidean isometric immersions with spectral Laplacian conditions, broadening understanding of minimal submanifolds in product spaces.

Contribution

It introduces new extensions of Takahashi's theorem to specific manifolds, analyzing spectral conditions of Laplacians in the context of minimal immersions.

Findings

Extended Takahashi's theorem to cylinders and tori.

Characterized minimal immersions via spectral Laplacian conditions.

Provided new insights into the geometry of minimal submanifolds in product spaces.

Abstract

In this note, we give natural extensions to cylinders and tori of a classical result due to T. Takahashi about minimal immersions into spheres. More precisely, we deal with Euclidean isometric immersions whose projections in R^N satisfy a spectral condition of their Laplacian.