Geometry of Harmonic Identity Maps
Aicha Benkartab, Ahmed Mohammed Cherif
TL;DR
This work investigates the harmonicity of identity maps under deformations of either the domain or codomain metric by $\tilde{g}= g- df\otimes df$ with $\|\operatorname{grad} f\|<1$. It derives explicit tension fields for the identity maps $\tilde{I}_c:(M,g)\to (M,\tilde{g})$ and $\tilde{I}_d:(M,\tilde{g})\to (M,g)$, showing $\tau(\tilde{I}_c) = -\frac{\Delta f}{1-\|\operatorname{grad} f\|^2} \operatorname{grad} f$ and $\tau(\tilde{I}_d) = \left[ \frac{\operatorname{Hess}_f(\operatorname{grad} f, \operatorname{grad} f)}{(1-\|\operatorname{grad} f\|^2)^2} + \frac{\Delta f}{1-\|\operatorname{grad} f\|^2} \right] \operatorname{grad} f$, which yield that $\tilde{I}_c$ is harmonic iff $f$ is harmonic on $(M,g)$ and that $\tilde{I}_d$ is harmonic under a Hessian/Laplace relation (affine $f$ gives harmonic $\tilde{I}_d$). The paper introduces a symmetric tensor $\chi = \|\operatorname{grad} f\|^{2} \operatorname{Hess}_f + (\Delta f)(g - df\otimes df)$ whose trace and evaluation on $\operatorname{grad} f$ characterize harmonicity of the two identity maps, and it provides a divergence formula for $\chi$ yielding rigidity results under nonnegative Ricci curvature and constant gradient norm. These results give precise criteria to construct new examples of harmonic identity maps, relate harmonicity to curvature and Hessian data, and establish sharp behavior on compact manifolds where harmonicity forces $f$ to be constant.
Abstract
An identity map $(M,g)\longrightarrow(M,g)$ is a harmonic from a Riemannian manifold $(M,g)$ onto itself. In this paper, we study the harmonicity of identity maps $(M,g)\longrightarrow(M,g-df\otimes df)$ and $(M,g-df\otimes df)\longrightarrow(M,g)$ where $f$ is a smooth function with gradient norm $<1$ on $(M,g)$. We construct new examples of identity harmonic maps. We define a symmetric tensor field on $M$ whose properties are related to the harmonicity of these identity maps.