Weak conformality of stable stationary maps for a functional related to conformality

TL;DR

This paper investigates the conformality properties of stable stationary maps between Riemannian manifolds, proving that under certain conditions, such maps must be weakly conformal, especially when mapping involving spheres of dimension five or higher.

Contribution

It establishes that stable C-stationary maps from or into spheres of dimension at least five are necessarily weakly conformal, extending understanding of conformality in geometric analysis.

Findings

Stable C-stationary maps on spheres of dimension ≥5 are weakly conformal.

Weak conformality is characterized by the vanishing of the tensor T_f.

The functional Φ measures deviation from conformality of the map.

Abstract

Let $(M, g)$, $(N, h)$ be compact Riemannian manifolds without boundary, and let $f$ be a smooth map from $M$ into $N$. We consider a covariant symmetric tensor $T_f$ $=$ ${\displaystyle f^*h - \frac{1}{m} |df|^2 g}$, where $f^*h$ denotes the pull-back metric of $h$ by $f$. The tensor $T_f$ vanishes if and only if the map $f$ is weakly conformal. The norm $|T_f|$ is a quantity which is a measure of conformality of $f$ at each point. We are concerned with maps which are critical points of the functional $\Phi (f)$ $=$ ${\displaystyle \int_M |T_f|^2dv_g}$. We call such maps {\it C-stationary maps}. Any conformal map or more generally any weakly conformal map is a C-stationary map. It is of interest to find when a C-stationary map is a (weakly) conformal map. In this paper we prove the following result. If $f$ is a stable C-stationary maps from the standard sphere ${\mathrm S}^m$ $(m \geq 5)$ or into the standard sphere ${\mathrm S}^n$ $(n \geq 5)$, then $f$ is a weakly conformal map.