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In this paper, we derive some local a priori estimates for the Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let g(t) be a smooth complete solution to the Ricci flow on R3, with the canonical Euclidean metric E as initial data, then g(t) is trivial, i.e. g(t) ≡ E.

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Let M be a compact Riemannian manifold, π : \tilde{M} → M be the universal covering and ω be a smooth 2-form on M with π^∗ω cohomologous to zero. Suppose the fundamental group π_1(M) satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we show that there exists a smooth 1-form η on \tilde{M} of linear (resp. bounded) growth such that π^∗ω = dη. As applications, we prove that on a compact Kahler manifold (M,ω) with π^∗ω cohomologous to zero, if π_1(M) is CAT(0) or automatic (resp. hyperbolic), then M is Kahler non-elliptic (resp. Kahler hyperbolic) and the Euler characteristic (−1)^{dim_RM/2χ(M) ≥ 0 (resp. >0).

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