We prove a general maximum principle at infinity for properly immersed minimal surfaces with boundary in R^3. An important corollary of this maximum principle at infinity is the existence of a fixed sized regular neighborhood for any properly embedded minimal surface of bounded curvature.

Let U be a connected non-singular quasi-projective variety and f : A → U a family of abelian varieties of dimension g. Suppose that the induced map U → Ag is generically finite and there is a compactification Y with complement S = Y \U a normal crossing divisor such that Ω1 Y (log S) is nef and ωY (S) is ample with respect to U. We characterize whether U is a Shimura variety by numerical data attached to the variation of Hodge structures, rather than by properties of the map U →Ag or by the existence of CM points. More precisely, we show that f : A → U is a Kuga fibre space, if and only if two conditions hold. First, each irreducible local subsystem V of R1f∗CA is either unitary or satisfies the Arakelov equality. Second, for each factor M in the universal cover of U whose tangent bundle behaves like that of a complex ball, an iterated Kodaira-Spencer map associated with V has minimal possible length in the direction of M. If in addition f : A → U is rigid, it is a connected Shimura subvariety of Ag of Hodge type.

Let be the scattering relation on a compact Riemannian manifold M with non-necessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let . be the length of that geodesic ray. We study the question of whether the metric g is uniquely determined, up to an isometry, by knowledge of 冃 and . restricted on some subset D. We allow possible conjugate points but we assume that the conormal bundle of the geodesics issued from D covers T M; and that those geodesics have no conjugate points. Under an additional topological assumption, we prove that 冃 and . restricted to D uniquely recover an isometric copy of g locally near generic metrics, and in particular, near real analytic ones.

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Let Sg be a closed surface of genus g and Mg be the moduli space of Sg endowed with the Weil–Petersson metric. In this article we investigate the Weil–Petersson curvatures of Mg for large genus g. First, we study the asymptotic behavior of the extremal Weil–Petersson holomorphic sectional curvatures at certain thick surfaces in Mg as g → ∞. Then we prove two curvature properties on the whole space Mg as g → ∞ in a probabilistic way.