Existence of rotationally symmetric embedded f-minimal tori

TL;DR

This paper extends the existence of rotationally symmetric embedded f-minimal tori to higher dimensions within a specific conformally flat metric influenced by a convex function.

Contribution

It generalizes Angenent's shrinking tori to minimal n-dimensional tori in a new class of conformally flat spaces with bounded convex functions.

Findings

Established existence of embedded f-minimal tori in higher dimensions.

Extended Angenent's classical shrinking tori to a broader geometric setting.

Demonstrated conditions on the convex function for the existence of such tori.

Abstract

We generalize Angenent's shrinking tori \cite{Angenent1992} to minimal $n$-dimensional tori embedded in $\mathbb{R}^{n+1}$ equipped with the metric $$g=e^{-\frac{f(\sum^{n+1}_{i=1}x_{i}^{2})}{2n}}\sum^{n+1}_{i=1}dx^{2}_{i},$$ where $f$ is a convex function and $f'$ is bounded above and below by positive constants.