We show that the dynamics of automorphisms on all projective complex manifolds X (of dimension 3, or of any dimension but assuming the Good Minimal Model Program or Mori’s Program) are canonically built up from the dynamics on just three types of projective complex manifolds: complex tori, weak Calabi-Yau manifolds and rationally connected manifolds. As a by-product, we confirm the conjecture of Guedj [20] for automorphisms on 3-dimensional projective manifolds, and also determine π1(X).

We state and prove a long-elusive relation between genus-one Gromov-Witten of a complete intersection and twisted Gromov Witten invariants of the ambient projective space. As shown in a previous paper, certain naturally arising cones of holomorphic vector bundle sections over the main component M 0 1,k(Pn, d) of the moduli space of stable genus-one holomorphic maps into Pn have a well-defined euler class. In this paper, we extend this result to moduli spaces of perturbed, in a restricted way, J-holomorphic maps. This extension is used to show that these cones are the correct genus-one analogues of the vector bundles relating genuszero Gromov-Witten invariants of a complete intersection to those of the ambient projective space. A relationship for higher-genus invariants is conjectured as well.

In \cite{rigidity}, Luo introduced a $\psi_{\lambda}$ edge invariant which turns out to be a coordinate of the Teichm\"uller space of a surface with boundary. And he proved that for $\lambda \geq 0$, the image of the Teichm\"uller space under $\psi_{\lambda}$ edge invariant coordinate is an open cell. In this paper we verify his conjecture that for $\lambda<0$, the image of the Teichm\"uller space is a bounded convex polytope.

In this paper we study constant mean curvature surfaces  in a product space, M2 × R, where M2 is a complete Riemannian manifold. We assume the angle function ν = hN, @ @t i does not change sign on . We classify these surfaces according to the infimum c() of the Gaussian curvature of the projection of . When H 6= 0 and c() ≥ 0, then  is a cylinder over a complete curve with curvature 2H. If H = 0 and c() ≥ 0, then  must be a vertical plane or  is a slice M2 ×{t}, or M2 ≡ R2 with the flat metric and  is a tilted plane (after possibly passing to a covering space). When c() < 0 and H > p −c()/2, then  is a vertical cylinder over a complete curve of M2 of constant geodesic curvature 2H. This result is optimal. We also prove a non-existence result concerning complete multigraphs in M2 × R, when c(M2) < 0.

In this article we introduce higher order Bergman functions for bounded complete Reinhardt domains in a variety with possi- bly isolated singularities. These Bergman functions are invariant under biholomorhic maps. We use Bergman functions to deter- mine all the biholomorhic maps between two such domains. As a result, we can construct an infinite family of numerical invari- ants from the Bergman functions for such  domains in An variety (x, y, z) ∈ C3 : xy = zn+1 . These infinite family of numerical invariants are actually a complete set of invariants for either the set of all bounded strictly pseudoconvex complete Reinhardt domain in An variety or the set of all bounded pseudoconvex complete Reinhardt domains with real analytic boundaries in An variety. In particular the moduli space of these domains in An variety is constructed explicitly as the image of this complete family of numerical invariants. It is well known that An variety is the quo- tient of cyclic group of order n + 1 on C2. We prove that the moduli space of bounded complete Reinhardt domains in An vari- ety coincides with the moduli space of the corresponding bounded complete Reinhardt domains in C2. Since our complete family of numerical invariants are computable, we have solved the biholo- morphically equivalent problem for large family of domains in C2.

We study Dirac structures associated with Manin pairs (d, g) and give a Dirac geometric approach to Hamiltonian spaces with D/G-valued moment maps, originally introduced by Alekseev and Kosmann-Schwarzbach [3] in terms of quasi-Poisson structures. We explain how these two distinct frameworks are related to each other, proving that they lead to isomorphic categories of Hamiltonian spaces. We stress the connection between the viewpoint of Dirac geometry and equivariant differential forms. The paper discusses various examples, including q-Hamiltonian spaces and Poisson-Lie group actions, explaining how presymplectic groupoids are related to the notion of “double” in each context.

In this paper, using connections between Clifford-Wolf isometries and Killing vector fields of constant length on a given Riemannian manifold, we classify simply connected Clifford-Wolf homogeneous Riemannian manifolds. We also get the classification of complete simply connected Riemannian manifolds with the Killing property defined and studied previously by J.E. D’Atri and H.K. Nickerson. In the last part of the paper we study properties of Clifford-Killing spaces, that is, real vector spaces of Killing vector fields of constant length, on odd-dimensional round spheres,and discuss numerous connections between these spaces and various classical objects.

For a compact Riemann surface X of positive genus, the space of sections of a certain theta bundle on moduli of bundles of rank r and level k admits a natural map to (the dual of) a similar space of sections of rank k and level r (the strange duality isomorphism). Both sides of the isomorphism carry projective connections as X varies in a family. We prove that this map is (projectively) flat.

In this paper we explicitly construct Moishezon twistor spaces on nCP2 for arbitrary n ≥ 2 which admit a holomorphic Caction. When n = 2, they coincide with Y. Poon’s twistor spaces. When n = 3, they coincide with the ones studied by the author in [14]. When n ≥ 4, they are new twistor spaces, to the best of the author’s knowledge. By investigating the anticanonical system, we show that our twistor spaces are bimeromorphic to conic bundles over certain rational surfaces. The latter surfaces can be regarded as orbit spaces of the C-action on the twistor spaces. Namely they are minitwistor spaces. We explicitly determine their defining equations in CP4. It turns out that the structure of the minitwistor space is independent of n. Further, we explicitly construct a CP2-bundle over the resolution of this surface, and provide an explicit defining equation of the conic bundles. It shows that the number of irreducible components of the discriminant locus for the conic bundles increases as n does. Thus our twistor spaces have a lot of similarities with the famous LeBrun twistor spaces, where the minitwistor space CP1 ×CP1 in LeBrun’s case is replaced by our minitwistor spaces found in [15].

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.

In this paper, we derive some local a priori estimates for the Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let g(t) be a smooth complete solution to the Ricci flow on R3, with the canonical Euclidean metric E as initial data, then g(t) is trivial, i.e. g(t) ≡ E.

Luttinger surgery is used to produce minimal symplectic 4- manifolds with small Euler characteristics. We construct a minimal symplectic 4-manifold which is homeomorphic but not diffeomorphic to CP2#3CP 2 , and which contains a genus two symplectic surface with trivial normal bundle and simply-connected complement. We also construct a minimal symplectic 4-manifold which is homeomorphic but not diffeomorphic to 3CP2#5CP 2 , and which contains two disjoint essential Lagrangian tori such that the complement of the union of the tori is simply-connected. These examples are used to construct minimal symplectic manifolds with Euler characteristic 6 and fundamental group Z, Z3, or Z/p ⊕ Z/q ⊕ Z/r for integers p, q, r. Given a group G presented with g generators and r relations, a symplectic 4-manifold with fundamental group G and Euler characteristic 10 + 6(g + r) is constructed.

It is classically known that complete flat (that is, zero Gaussian curvature) surfaces in Euclidean 3-space R3 are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface f admits singularities but its Gauss map  is globally defined on the surface and can be smoothly extended across the singular set, f is called a frontal. In addition, if the pair (f, ) defines an immersion into R3×S2, f is called a front. A front f is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the caustic (i.e. focal surface) of a flat front f are also flat fronts. In this paper, we generalize the classical notion of completeness to flat fronts, and give a representation formula for a flat front which has a non-empty compact singular set and whose ends are all immersed and complete. As an application, we show that such a flat front has properly embedded ends if and only if its Gauss map image is a convex curve. Moreover, we show the existence of at least four singular points other than cuspidal edges on such a flat front with embedded ends, which is a variant of the classical four vertex theorem for convex plane curves.

By using variational method and under generic condition, we show that Arnold diffusion exists in a priori hyperbolic and timeperiodic Hamiltonian systems with multiple degrees of freedom.

In [10], R. Hamilton established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator. We show that this inequality holds under the weaker condition that M × R2 has nonnegative isotropic curvature.

Given an SO(3)-bundle with connection, the associated twosphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov [34]. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to construct symplectic six-manifolds with c1 = 0 which are never K¨ahler; e.g., we produce such manifolds with b1 = 0 = b3 and also with c2 · [ω] < 0, answering questions posed by Smith–Thomas–Yau [37]. Examples come from Riemannian geometry, via the Levi–Civita connection on +. The underlying six-manifold is then the twistor space and often the symplectic structure tames the Eells–Salamon twistor almost complex structure. Our second aim is to exploit this to deduce new results about minimal surfaces: if a certain curvature inequality holds, it follows that the space of minimal surfaces (with fixed topological invariants) is compactifiable; the minimal surfaces must also satisfy an adjunction inequality, unifying and generalising results of Chen–Tian [6]. One metric satisfying the curvature inequality is hyperbolic four-space H4. Our final aim is to show that the corresponding symplectic manifold is symplectomorphic to the small resolution of the conifold xw −yz = 0 in C4. We explain how this fits into a hyperbolic description of the conifold transition, with isometries of H4 acting symplectomorphically on the resolution and isometries of H3 acting biholomorphically on the smoothing.

The diameter of a disc filling a loop in the universal covering of a Riemannian manifold M may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram
filling a word that represents the identity in a finitely presented group 􀀀 can either be measured intrinsically in the 1–skeleton of
or extrinsically in the Cayley graph of 􀀀. We construct the first examples of closed manifolds M and finitely presented groups 􀀀 = π1M for which this choice — intrinsic versus extrinsic — gives rise to qualitatively different min–diameter filling functions.

Following Matveev, a k-normal surface in a triangulated 3-manifold is a generalization of both normal and (octagonal) al- most normal surfaces. Using spines, complexity, and Turaev-Viro invariants of 3-manifolds, we prove the following results: a minimal triangulation of a closed irreducible or a boundedhyperbolic 3-manifold contains no non-trivial k-normal sphere; every triangulation of a closed manifold with at least 2 tetra-hedra contains some non-trivial normal surface; every manifold with boundary has only finitely many triangulations without non-trivial normal surfaces. Here, triangulations of bounded manifolds are actually ideal triangulations. We also calculate the number of normal surfaces of nonnegative Euler characteristics which are contained in the conjecturally minimal triangulations of all lens spaces Lp,q.

Let w be an Abelian differential on a compact Riemann surface of genus g ≥ 1. Then |w|2 defines a flat metric with conical singularities and trivial holonomy on the Riemann surface. We obtain an explicit holomorphic factorization formula for the ζ-regularized determinant of the Laplacian in the metric |w|2, generalizing the classical Ray-Singer result in g = 1.

The chains studied in this paper generalize Chern–Moser chains for CR structures. They form a distinguished family of one dimensional submanifolds in manifolds endowed with a parabolic contact structure. Both the parabolic contact structure and the system of chains can be equivalently encoded as Cartan geometries (of different types). The aim of this paper is to study the relation between these two Cartan geometries for Lagrangean contact structures and partially integrable almost CR structures. We develop a general method for extending Cartan geometries which generalizes the Cartan geometry interpretation of Fefferman’s construction of a conformal structure associated to a CR structure. For the two structures in question, we show that the Cartan geometry associated to the family of chains can be obtained in that way if and only if the original parabolic contact structure is torsion free. In particular, the procedure works exactly on the subclass of (integrable) CR structures. This tight relation between the two Cartan geometries leads to an explicit description of the Cartan curvature associated to the family of chains. On the one hand, this shows that the homogeneous models for the two parabolic contact structures give rise to examples of non–flat path geometries with large automorphism groups. On the other hand, we show that one may (almost) reconstruct the underlying torsion free parabolic contact structure from the Cartan curvature associated to the chains. In particular, this leads to a very conceptual proof of the fact that chain preserving contact diffeomorphisms are either isomorphisms or antiisomorphisms of parabolic contact structures.

We consider the zeros on the boundary @­ of a Neumann eigen- function '¸j of a real analytic plane domain ­. We prove that the number of its boundary zeros is O(¸j) where ¡¢'¸j = ¸2j '¸j . We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is O(¸j ). It follows that the number of nodal lines of '¸j (components of the nodal set) which touch the boundary is of order ¸j . This upper bound is of the same order of magnitude as the length of the total nodal line, but is the square root of the Courant bound on the number of nodal components in the interior. More generally, the results are proved for piecewise analytic domains.

The limiting behavior of the normalized KÄahler-Ricci °ow for manifolds with positive ¯rst Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi Kenergy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabuchi Kenergy is bounded from below and if the lowest positive eigenvalue of the ¹@y ¹@ operator on smooth vector ¯elds is bounded away from0 along the °ow, then the metrics converge exponentially fast in C1 to a KÄahler-Einstein metric.

We prove that the rotation in time T of a trajectory of a K-Lipschitz vector ¯eld in Rn around a given point (stationary or non-stationary) is bounded by A + BKT with A;B absolute constants. In particular, trajectories of a Lipschitz vector ¯eld in ¯nite time cannot have an in¯nite rotation around a given point (while trajectories of a C1 vector ¯eld may have an in¯- nite rotation around a straight line in ¯nite time). The bound above extends to the mutual rotation of two trajectories (for the time intervals T and T0, respectively) of a K-Lipschitz vector ¯eld in R3: this rotation is bounded from above by the quantity CK min(T; T0) + DK2TT0.

We prove the factoriality of a nodal hypersurface in P4 of degree d that has at most 2(d ¡ 1)2=3 singular points, and we prove the factoriality of a double cover of P3 branched over a nodal surface of degree 2r having less than (2r ¡ 1)r singular points.

We prove that Mark Gross’ [8] topological Calabi-Yau compactifications can be made into symplectic compactifications. To prove this we develop a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. This produces pairs of compact symplectic 6-manifolds homeomorphic to mirror pairs of Calabi-Yau 3-folds together with Lagrangian fibrations whose underlying integral affine structures are dual.