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In this paper, we consider a manifold evolving by a general geometric flow and study parabolic equationWe establish space-time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this equation. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. Finally, as applications, we give gradient estimates of some specific parabolic equations.

We establish a Schwarz lemma for $V$-harmonic maps of generalised dilatation between Riemannian manifolds. We apply the result to obtain corresponding results for Weyl harmonic maps of generalised dilatation from conformal Weyl manifolds to Riemannian manifolds and holomorphic maps from almost Hermitian manifolds to quasi-Kähler and almost Kähler manifolds.

In this paper, we prove an isolation theorem for holomorphic and anti-holomorphic maps from a compact conformal semi-Kähler manifold with or without boundary to an almost Kähler manifold with constrained energy density, which extends the results of Gauchman and Glazebrook. At the same time, as an analogy of a theorem of Ilias and Shouman, the corresponding results are improved.

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