# A NONEXISTENCE THEOREM FOR STABLE EXPONENTIALLY HARMONIC MAPS

• Koh, Sung-Eun (Department of Mathematics, Konkuk University, Seoul 133-701)
• Published : 1995.08.01

#### Abstract

Let M and N be compact Riemannian manifolds and $f : M \to N$ be a smooth map. Following J. Eells, f is exponentially harmonic if it represents a critical point of the exponential energy integral $$E(f) = \int_{M} exp(\left\\mid df \right\\mid^2) dM$$ where $(\left\ df$\mid$\right\$\mid$^2$ is the energy density defined as $\sum_{i=1}^{m} \left\$\mid$df(e_i) \right\$\mid$^2$, m = dimM, for orthonormal frame $e_i$ of M. The Euler- Lagrange equation of the exponential energy functional E can be written $$exp(\left\\mid df \right\\mid^2)(\tau(f) + df(\nabla\left\\mid df \right\\mid^2)) = 0$$ where $\tau(f)$ is the tension field along f. Hence, if the energy density is constant, every harmonic map is exponentially harmonic and vice versa.