Shrinking doughnuts via variational methods

TL;DR

This paper presents a novel variational approach and modified curvature flow to prove the existence of self-shrinking tori under mean curvature flow, providing new bounds on their weighted energy.

Contribution

It introduces an alternative proof for self-shrinking tori existence using variational methods and curvature flow modifications, with implications for energy bounds.

Findings

Existence of self-shrinking tori under mean curvature flow confirmed.

An upper bound for the weighted energy of shrinking doughnuts established.

Provides an alternative proof method compared to traditional approaches.

Abstract

We use variational methods and a modified curvature flow to give an alternative proof of the existence of a self-shrinking torus under mean curvature flow. As a consequence of the proof, we establish an upper bound for the weighted energy of our shrinking doughnuts.