The Existence of Non-Equivariant Gromov Tori

In this talk, I will address the following question: if a flat torus ${\mathbb{T}}^n$ is isometrically and minimally embedded into a sphere ${\mathbb{S}}^N$, must its translation group extend to the isometry group of the ambient sphere? As shown by Robert Bryant, for $n=2$ the answer is positive. Also, Ying Lu, Peng Wang, and Zhenxiao Xie recently demonstrated that for immersions, the answer is negative when $n\ge 3$. This problem is deeply tied to the work of Mikhail Gromov and Anton Petrunin concerning optimal curvature bounds. Petrunin proved that any immersion of a torus into a unit ball must have a maximum normal curvature of at least $\sqrt{\frac{3n}{n+2}}$. This bound is attained, for example, by families of tori constructed by Gromov. We call the tori that attain this optimal bound "Gromov tori". During the talk, I will first demonstrate that any Gromov torus is intrinsically flat, lies within a sphere, and is minimal inside it. We will then establish the necessary and sufficient conditions for defining these tori. Finally, I will present the main result of our recent work: for dimensions $n\ge 3$, there exists a non-equivariant embedded Gromov torus, which answers the question above.
RECORDING of MEETING