We investigate minimal extension sheaves on arbitrary (possibly non-rational) fans as an approach toward a combinatorial "virtual'' intersection cohomology. These are flabby sheaves of graded modules over a sheaf of polynomial rings, satisfying three relatively simple axioms that characterize the equivariant intersection cohomology sheaves on toric varieties. As in "classical'' intersection cohomology, minimal extension sheaves are models for the pure objects of a "perverse category"; a Decomposition Theorem holds. The analysis of the step from equivariant to non-equivariant intersection cohomology of toric varieties leads us to investigate "quasi-convex" fans (generalizing fans with convex or "co-convex" support), where our approach yields a meaningful virtual intersection cohomology. We characterize such fans by a topological condition and prove a version of Stanley's "Local-Global" formula relating the global intersection Poincaré polynomial to local data. Virtual intersection cohomology of quasi-convex fans is shown to satisfy Poincaré duality. To describe the local data in terms of the global data for lower-dimensional complete polytopal fans as in the rational case, one needs a "Hard Lefschetz" type result. It requires a vanishing condition that is valid for rational cones, but has not yet been proven in the general case.

We consider a Banach space of finite-dimensional-Hilbert-space-valued functions on a sigma-finite measure space. The norm of the function space is assumed to satisfy some suitable conditions. Then we prove a pointwise local ergodic theorem for a $(C_{0})$-semigroup of linear contractions on the function space, under an additional norm condition for operators of the semigroup. Our result extends Baxter and Chacon's local ergodic theorem for scalar-valued functions.

We compute the Petersson scalar product of certain Poincaré series introduced in our previous paper against a Siegel cusp form and show that it can be written as a certain averaged cycle integral. This generalizes earlier work by Katok, Zagier and the first named author in the case of genus 1.

We compute the first variation of the functional that assigns each unit vector field the volume of its image in the unit tangent bundle. It is shown that critical points are exactly those vector fields that determine a minimal immersion. We also find a necessary and sufficient condition that a vector field, defined in an open manifold, must fulfill to be minimal, and obtain a simpler equivalent condition when the vector field is Killing. The condition is fulfilled, in particular, by the characteristic vector field of a Sasakian manifold and by Hopf vector fields on spheres.

We present sufficient conditions of extending a meromorphic function which is defined outside an analytic compact curve in a complex surface. The function we deal with is a first integral for a holomorphic foliation in the whole surface. The key to extension is the study of singularities of the foliation on the complex curve.

We study a free boundary value problem of the Euler system arising in the inviscid steady supersonic flow past a symmetric curved cone. The existence and stability of piesewise smooth weak entropy solutions was established, provided the cone is a small perturbation of its tangential cone with a vertex angle less than a given value determined by the parameters of the coming flow. Since the change of the entropy of the flow is also considered, the result in this paper gives a more precise description than previous ones on such problems.

It is well-known that the only minimal Lagrangian submanifolds of constant sectional curvature $c$ in a Riemannian complex space form of constant holomorphic sectional curvature $4c$ are the totally geodesic ones. In this paper we investigate minimal Lagrangian Lorentzian submanifolds of constant sectional curvature $c$ in Lorentzian complex space form of constant holomorphic sectional curvature $4c$. We prove that the situation in the Lorentzian case is quite different from the Riemannian case. Several existence and classification theorems in this respect are obtained. Some explicit expression of flat minimal Lagrangian submanifolds in flat complex Lorentzian space form are also presented.

For a CMC immersion from a two-dimensional compact smooth manifold with boundary into the Euclidean three-space, we give sufficient conditions under which it has a CMC deformation fixing the boundary. Moreover, we give a criterion of the stability for CMC immersions. Both of these are achieved by using the properties of eigenvalues and eigenfunctions of an eigenvalue problem associated to the second variation of the area functional. In a certain special case, by combining these results, we obtain a 'visible' way of judging the stability.