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We study the evolution by mean curvature of a smooth n–dimensional surfaceM Rn+1, compact and with positive mean curvature. We first prove an estimate on the negative part of the scalar curvature of the surface. Then we apply this result to study the formation of singularities by rescaling techniques, showing that there exists a sequence of rescaled flows converging to a smooth limit flow of surfaces with nonnegative scalar curvature. This gives a classification of the possible singular behaviour for mean convex surfaces in the case n = 2

We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the mth mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a suitable pinching condition, the solution exists for all times and converges to a round sphere. Mathematics Subject Classification (2000)53C44 · 35K55 · 58J35 · 35B40