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We prove that the image of period map is algebraic, as conjectured by Griffiths.

In this paper, we study the global well-posedness of the 2D compressible NavierStokes equations with large initial data and vacuum. It is proved that if the shear viscosity<i></i> is a positive constant and the bulk viscosity <i></i> is the power function of the density, that is, <i></i>(<i></i>) = <i></i> ^<i></i> with <i></i> &gt; 3, then the 2D compressible NavierStokes equations with the periodic boundary conditions on the torus T 2 admit a unique global classical solution (<i>, u</i>) which may contain vacuums in an open set of T 2 . Note that the initial data can be arbitrarily large to contain vacuum states.

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We prove that a compact Riemannian 3-symmetric space is globally symmetric if every Jacobi field along a geodesic vanishing at two points is the restriction to that geodesic of a Killing field induced by the isotropy action or, in particular, if the isotropy action is variationally complete.