In this paper we show that if two strictly convex, compact real projective manifolds have the same marked length spectrum with respect to the Hilbert metric, then they are projectively equivalent. This is a rigidity for Finsler metric with a special geometric structure. Furthermore we prove an analogue of a Hitchin's conjecture for hyperbolic 3-manifolds, namely the deformation space of convex real projective structures on a compact hyperbolic 3-manifold M is a component in the moduli space of PGL(4,ℍ)-representations of π₁(M).

We show that the existence of a maximal embedded tube in a hyperbolic n-manifold implies the existence of a certain conical region. One application is to establish a lower bound on the volume of the region outside the tube, thereby improving estimates on volume and estimates on lengths of geodesics in small volume hyperbolic 3-manifolds. We also provide new bounds on the injectivity radius and diameter of an n-manifold.

In this paper we investigate connections between minimal Lagrangian submanifolds and holomorphic vector fields in Kähler manifolds. Our main result is: Let M²ⁿ (n ≥ 2) be a Kähler-Einstein manifold with positive scalar curvature with an effective, structure-preserving action by an n-torus Tⁿ. Then precisely one regular orbit L of the Tⁿ-action is a minimal Lagrangian submanifold of M. Moreover there is an (n − 1)-torus Tⁿ⁻¹ ⊂ Tⁿ and a sequence of non-flat immersed minimal Lagrangian tori L_k in M such that all L_k are invariant under Tⁿ⁻¹ and L_k locally converge to L (in particular the supremum of the sectional curvatures of L_k and the distance between L and L_k go to 0 as k ↦ ∞). This result is new even for M = ℂPⁿ for n ≥ 3.

Let $M$ be a compact, connected, orientable, hyperbolic 3-manifold whose boundary is a torus and which contains an essential closed surface $S$. It is conjectured that 5 is an upper bound for the distance between two slopes on $\partial M$ whose associated fillings are not hyperbolic manifolds. In this paper we verify the conjecture when the first Betti number of $M$ is at least 2 by showing that given a pseudo-Anosov mapping class $f$ of a surface and an essential simple closed curve $\gamma$ in the surface, then 5 is an upper bound for the diameter of the set of integers $n$ for which the composition of $f$ with the $n^{th}$ power of a Dehn twist along $\gamma$ is not pseudo-Anosov. This sharpens an inequality of Albert Fathi. For large manifolds $M$ of first Betti number 1 we obtain partial results. Set $$ \mathcal C(S) = {\rm{slopes} \, r \, | \, ker(\pi_1 (S) \to \pi_1 (M(r))) \neq {1} } $$ A singular slope for $S$ is a slope $r_0 \in \mathcal{C} (S)$ such that any other slope in $\mathcal{C}S$ is at most distance 1 from $r_0$. We prove that the distance between two exceptional filling slopes is at most 5 if either (i) there is a closed essential surface $S$ with $ \mathcal{C}S$ finite, or (ii) there are singular slopes $r_1 \neq r_2$ for closed essential surfaces $S_1, S_2$ in $M$.

In this paper, we proved that if a singular manifold satisfies a weak mean value property for positive subharmonic functions then one can derive an oscillation bound for bounded holomorphic functions. Moreover, if we further assume that the volume decays at most polynomially at a singular point, then we obtain a Hölder estimate of the holomorphic function at that point. In a similar spirit, we also established a continuity estimate for bounded harmonic functions, with a finite dimensional exception, at a singular point of a manifold satisfying a weak mean value property and a polynolmial volume decay condition.

Let M be a closed oriented smooth 4-manifold admitting symplectic structures. If M is minimal and has b⁺ = 1, we prove that there is a unique symplectic canonical class up to sign, and any real second cohomology class of positive square is represented by symplectic forms. Similar results hold when M is not minimal.