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.