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
, 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
, 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
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.