In the general, this article addresses the nature of, and relation between, canonical and log-canonical singularities of foliations by curves. The case of equivariance under a finite group action has several peculiarities, amongst which are examples from dimension 3 onward where no birational modification can be both Gorenstein and log-canonical. The article in the specific, i.e. a functorial resolution algorithm for foliated 3-folds in the 2-category of Deligne-Mumford champs, is thus not only optimal but natural.

Inspired by the Gromov-Hausdorff distance, we define a new notion called the intrinsic flat distance between oriented m dimensional Riemannian manifolds with boundary by isometrically embedding the manifolds into a common metric space, measuring the flat distance between them and taking an infimum over all isometric embeddings and all common metric spaces. This is made rigorous by applying Ambrosio-Kirchheim’s extension of FedererFleming’s notion of integral currents to arbitrary metric spaces. We prove the intrinsic flat distance between two compact oriented Riemannian manifolds is zero iff they have an orientation preserving isometry between them. Using the theory of AmbrosioKirchheim, we study converging sequences of manifolds and their limits, which are in a class of metric spaces that we call integral current spaces. We describe the properties of such spaces including the fact that they are countably Hm rectifiable spaces and present numerous examples.

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.

We prove that given a three manifold with an arbitrary metric $(M^3, g)$ of positive Ricci curvature, there exists a sweepout of $M$ by surfaces of genus $\leq 3$ and areas bounded by $C vol(M^3, g)^{2/3}$. We use this result to construct a sweepout of $M$ by 1-cycles of length at most $C vol(M^3, g)^{1/3}$ and prove a systolic inequality for all $M \neq S^3$. The sweepout of surfaces is generated from a min-max minimal surface. If further assuming a positive scalar curvature lower bound, we can get a diameter upper bound for the min-max surface.

Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform: in its isotopy class to some" canonical" embedded surface. From the point of view of geometry, a natural" canonical" surface will be the extremal surface of some functional defined on the space of embedded surfaces. The simplest functional is the area functional. The extremal surface of the area functional is called the minimal surface. Such minimal surfaces were used extensively by Meeks-Yau [MY21 in studying group actions on three dimensional manifolds.