Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere on $N$ is a harmonic map from $({ℝ}^m,e^{{\left|x\right|}^2/2(m-2)}/d{s}_0^2)$ to $N$ ($m\ge 3$) with finite energy ([LnW]). Here $ds{2}_0$ is the Euclidean metric in ${ℝ}^m$. Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$. We also derive gradient estimates and Liouville theorems for positive...

We introduce the concept of conserved current variationally associated with locally variational invariant field equations. The invariance of the variation of the corresponding local presentation is a sufficient condition for the current beeing variationally equivalent to a global one. The case of a Chern-Simons theory is worked out and a global current is variationally associated with a Chern-Simons local Lagrangian.

Let $G:{ℂ}^n\times {ℂ}^r\to ℂ$ be a holomorphic family of functions. If $\Lambda \subset {ℂ}^n\times {ℂ}^r$, ${\pi }_r:{ℂ}^n\times {ℂ}^r\to {ℂ}^r$ is an analytic variety then   $Q_{\Lambda }\left(G\right)=(x,u)\in {ℂ}^n\times {ℂ}^r:G(·,u)hasacriticalpointin\Lambda \cap {\pi }_r^{-1}\left(u\right)$is a natural generalization of the bifurcation variety of G. We investigate the local structure of $Q_{\Lambda }\left(G\right)$ for locally trivial deformations of $\Lambda ₀={\pi }_r^{-1}\left(0\right)$. In particular, we construct an algorithm for determining logarithmic stratifications provided G is versal.