We consider n-hypersurfaces Σ_j with interior E_j whose mean curvature are given by the trace of an ambient Sobolev function u_j ∊ W^1,p(ℝⁿ⁺¹)

(0.1) \bar H_Σj = u_jν_Ej on Σ_j,

where ν_Ej denotes the inner normal of Σ_j. We investigate (0.1) when Σ_j → Σ weakly as varifolds and prove that Σ is an integral n-varifold with bounded first variation which still satisfies (0.1) for u_j → u, E_j → E. p has to satisfy

p > 1/2 (n + 1)

and p ≥ 4/3 if n = 1. The difficulty is that in the limit several layers can meet at Σ which creates cancellations of the mean curvature.

Let F be a closed surface. If i, i′ : F → ℝ³ are two regularly homotopic generic immersions, then it has been shown in [5] that all generic regular homotopies between i and i′ have the same number mod 2 of quadruple points. We denote this number by Q(i, i′) ∊ Z/2. For F orientable we show that for any generic immersion i : F → ℝ³ and any diffeomorphism h : F → F such that i and i º h are regularly homotopic,

Q(i, i º h) = (rank(h_* − Id) + (n + 1)∊(h)) mod 2,

where h_* is the map induced by h on H₁(F, ℤ/2), n is the genus of F and ∊(h) is 0 or 1 according to whether h is orientation preserving or reversing, respectively.

We then give an explicit formula for Q(e, e′) for any two regularly homotopic embeddings e, e′ : F → ℝ³. The formula is in terms of homological data extracted from the two embeddings.

For isometric actions on flat Lorentz (2+1)-space whose linear part is a purely hyperbolic subgroup of O(2, 1), Margulis defined a marked signed Lorentzian length spectrum invariant closely related to properness and freeness of the action. In this paper we show that, for fixed linear part, this invariant completely determines the conjugacy class of the action. We also extend this result to groups containing parabolics.

We give a new construction of Lie groupoids which is particularly well adapted to the generalization of holonomy groupoids to singular foliations. Given a family of local Lie groupoids on open sets of a smooth manifold M, satisfying some hypothesis, we construct a Lie groupoid which contains the whole family. This construction involves a new way of considering (local) Morita equivalences, not only as equivalence relations but also as generalized isomorphisms. In particular we prove that almost injective Lie algebroids are integrable.

In this paper, we studied complete manifolds whose spectrum of the Laplacian has a positive lower bound. In particular, if the Ricci curvature is bounded from below by some negative multiple of the lower bound of the spectrum, then we established a splitting type theorem. Moreover, if this assumption on the Ricci curvature is only valid outside a compact subset, then the manifold must have only finitely many ends with infinite volume. Similar type theorems are also obtained for complete Kähler manifolds.

We give two versions of relative hyperbolization. We use the first version to prove that if (each component of) a closed manifold M is aspherical and if M is a boundary, then it is the boundary of an aspherical manifold.