Abstract. The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, and tame geometry. This paper provides alternative characterizations of this type of inequality for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In the framework of metric spaces, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka-Lojasiewicz inequality) is related to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by −∂f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the KurdykaLojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines -a concept linked to the location of the less steepest points at the level sets of f -and integrability conditions are given. In the convex case these results are significantly reinforced, allowing us in particular to establish a kind of asymptotic equivalence for discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C 2 function in R 2 is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka-Lojasiewicz inequality.

In this paper, we study the existence and uniqueness of solutions to Jenkins-Serrin type problems on domains in a Riemannian surface: infinite boundary data are allowed. In the case of unbounded domains, the study is focused on the hyperbolic plane. A counterexample to uniqueness is given for entire minimal graphs over H 2 .

Abstract. In this paper, we study closed embedded minimal hypersurfaces in a Riemannian (n + 1)-manifold (2 ≤ n ≤ 6) that minimize area among such hypersurfaces. We show they exist and arise either by minimization techniques or by min-max methods : they have index at most 1. We apply this to obtain a lower area bound for such minimal surfaces in some hyperbolic 3-manifolds.

In this note, we explain how a mistake made in [2] can be corrected. Actually this mistake appears in the proof of Proposition 8 (the second maximum principle) and was brought to our attention by A. Song [4]. Let us notice that unfortunately, we did not find an alternative proof of this proposition but we found an alternative proposition. The new proposition does not change the subsequent applications we made of the original proposition. At the end of the note we explain which modifications should be done where the original Proposition 8 is applied.The difference with the original Proposition 8 is that here we have to assume a control on the index of the minimal surface Σ.The new proposition. Let (T, ds 2 T ) be a flat 2 torus of diameter 1 and consider M = T×[− 1 2 , +∞) endowed with the metric g = e −2h(t) Λ 2 ds 2 T +dt 2 ; here Λ is a positive constant and h satisfies the following assumptions:•The main example is h(t) = t. We denote by T s = T × {s}.Proposition 1. Let i 0 ∈ N. There is a Λ 0 such that the following is true. For any Λ ≤ Λ 0 and any compact embedded minimal surface Σ ⊂ M with ∂Σ ⊂ T − 1 2 and index less than i 0 , we have Σ ⊂ T × [− 1 2 , 0]Proof. If the proposition is not true there is a sequence Λ n → 0 and Σ n a minimal surface in M with index less than i 0 , boundary in T − 1 2 and the maximum of the function t on Σ n is t n > 0.In order to study this sequence, we translate in the t direction by −t n and make a homothety by λ n = e h(tn) Λn → +∞. We thus obtain a compact minimal surface S n in T × [−( 1 2 + t n )λ n , 0] endowed with the metric g n = e −2hn(t) ds 2 T + dt 2 (where h n (t) = h( t λn + t n ) − h(t n )) with boundary in T −( 1 2 +tn)λn and containing a point in T 0 . We notice that h n → 0 uniformly on any compact. So the ambient space smoothly converges to X = T × R − endowed with the flat metric ds 2 T × dt 2 . Let ε be positive. Since S n has index at most i 0 , there is a set E n (ε) of at most i 0 points in S n such that, for any p ∈ X with d(p, E n (ε)) > 2ε, 1