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Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if Mn is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition R0 > nKmax, where n 2 ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We also provide a partial answer to Yau’s conjecture on the pinching theorem. Moreover, we prove that if Mn(n  3) is a compact manifold whose (n . 2)-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition Ric(n.2) min > n(n .2)R0, where n 2 ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker, and the authors.

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