From any 4-dimensional oriented handlebody X without 3- and 4-handles and with b2  1, we construct arbitrary many compact Stein 4-manifolds that are mutually homeomorphic but not diffeomorphic to each other, so that their topological invariants (their fundamental groups, homology groups, boundary homology groups, and intersection forms) coincide with those of X. We also discuss the induced contact structures on their boundaries. Furthermore, for any smooth 4-manifold pair (Z, Y ) such that the complement Z − int Y is a handlebody without 3- and 4-handles and with b2  1, we construct arbitrary many exotic embeddings of a compact 4-manifold Y 0 into Z, such that Y 0 has the same topological invariants as Y .

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We propose a geometric inequality for two-dimensional spacelike surfaces in the Schwarzschild spacetime. This inequality implies the Penrose inequality for collapsing dust shells in general relativity, as proposed by Penrose and Gibbons. We prove that the inequality holds in several important cases.

We study the action of the group of contact diffeomorphisms on CR deformations of compact three dimensional CR manifolds. Using anisotropic function spaces and an anisotropic structure on the space of contact diffeomorphisms, we establish the existence of local transverse slices to the action of the contact diffeomorphism group in the neighborhood of a fixed embeddable strongly pseudoconvex CR structure.

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