Existence of rotationally symmetric embedded f-minimal tori

Abstract

We generalize Angenent's shrinking torus \cite{Angenent1992} by proving the existence of an embedded $n$-dimensional $f$-minimal hypersurface diffeomorphic to $S^1\times S^{n-1}$ in ${\mathbb{R}}^{n+1}$ equipped with the metric $$d{s}^2=e^{-\frac{f(|x{|}^2)}{2n}}\sum _{i=1}^{n+1}d{x}_i^2,$$ where $f$ is a convex function of $|x{|}^2$ satisfying $m\le f^{\prime }\le M$ for some positive constants $m$ and $M$.

14 pages