WE CONSIDER Haken manifolds M. That is three manifolds which are compact, irreducible, orientable, and sufficiently 1arge. S 8M may or may not be empty. This class of 3-manifolds shares many properties with closed surfaces (# S2 or RP2). On an algebraic level, both are examples of K (T, 1)s. More geometrically the hierarchy structure of a Haken manifold [9] is analogous to a sequence of cuts turning a surface into a disk. With the aid of minimal surfaces we are able to find a sufficiently natural hierarchy to establish a new similarity between Haken manifolds and surfaces. Our main theorem is:

In this paper we study the prescribed centroaffine curvature problem in the Euclidean space R n+ 1. This problem is equivalent to solving a Monge鈥揂mp猫re equation on the unit sphere. It corresponds to the critical case of the Blaschke鈥揝antal贸 inequality. By approximation from the subcritical case, and using an obstruction condition and a blow-up analysis, we obtain sufficient conditions for the a priori estimates, and the existence of solutions up to a Lagrange multiplier.

Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Yamabe flow introduced by Luo (Commun Contemp Math 6(5):765–780, 2004). To handle the singularies along the combinatorial Calabi flow, we do surgery on the flow by flipping. Using the discrete conformal theory established in Gu et al. (J Differ Geom 109(3):431–466, 2018; J Differ Geom 109(2):223–256, 2018), we prove that for any initial Euclidean or hyperbolic polyhedral metric on a closed surface, the combinatorial Calabi flow with surgery exists for all time and converges exponentially fast after finite number of surgeries. The convergence is independent of the combinatorial structure of the initial triangulation on the surface.

We prove that a general miraculous cancellation formula, the divisibility of certain characteristic numbers, and some other topological results related to the generalized Rochlin invariant, the -invariant and the holonomies of certain determinant line bundles, are consequences of the modular invariance of elliptic operators on loop space.

For any smooth Riemannian metric on an (n+1)-dimensional compact manifold with boundary (M,∂M) where 3≤(n+1)≤7, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates to prove that for almost every (in the C-infinity Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M. If ∂M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.