arXiv research

Defines products for fibered corners manifolds, generalizing resolutions.

We investigate the Kahler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kahler metrics. This strengthens previous…

Given a Kähler fiber space $p:X\to Y$ whose generic fiber is of general type, we prove that the fiberwise singular Kähler-Einstein metric induces a semipositively curved metric on the relative canonical bundle $K_{X/Y}$ of $p$. We also propose a conjectural generalization of this result for relative twisted Kähler-Eins…

Study variations of metrics on Riemannian submersions to preserve fiber geometry.

We consider a homogeneous fibration $G/L \to G/K$, with symmetric fiber and base, where $G$ is a compact connected semisimple Lie group and $L$ has maximal rank in $G$. We suppose the base space $G/K$ is isotropy irreducible and the fiber $K/L$ is simply connected. We investigate the existence of $G$-invariant Einstein…

Enhances geodesic fiber tracking in white matter using modified metrics and tensor data.

In this work we study the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for such a metric to be Einstein in terms …

Surjectivity of Cannon-Thurston map proven for metric graph bundles.

Study variations of Riemannian submersions to maintain geodesic fibers and positive curvatures.

Counterexample disproves conjecture on flat metrics and fiber bundles.

New techniques compute $L^2$-cohomology of quasi-fibered metrics.

Theorems prove upper bounds for foliations on closed Alexandrov spaces.

Let $P\to M$ be a principal bundle. Consider a sequence of metrics on $P$ obtained by re-scaling the fibers to points. The Gromov-Hausdorff limit of the tangent bundles over these principal bundles with their Sasaki metric is seen herein to be a locally trivial fiber bundle containing the tangent space to the base as a…

This paper is a sequel to arXiv:1108.0967. We further study Gromov-Hausdorff collapsing limits of Ricci-flat Kähler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as arXiv:1108.0967, if the dimension of the base manifold is one, the limit metric space is homeomorphic to the bas…

The paper proves non-existence of positive scalar curvature on certain fiber bundles.

Proves estimates for Calabi-Yau metrics as Kahler classes shrink.

We study adiabatic limits of Ricci-flat Kahler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Ampere equation, we show that the Ricci-flat metrics collapse (away f…

In this note we prove that any closed graph manifold admitting a metric of non-positive sectional curvature (NPC-metric) has a finite cover, which is fibered over the circle. An explicit criterion to have a finite cover, which is fibered over the circle, is presented for the graph manifolds of certain class.

Study shows how certain curved bundles reduce their structure group.

For any elliptic K3 surface $\mathfrak{F}: \mathcal{K} \rightarrow \mathbb{P}^1$, we construct a family of collapsing Ricci-flat Kähler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov-Hausdorff limit to $\mathbb{P}^1$ equipped with the McLean metric. There are well-known e…

We consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e. polyhomogeneous …

Construct Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.

Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the fam…

New gradient Ricci solitons found via fiber bundles.

We prove that the indices of fibered-cusp and $d$-Dirac operators on a spin manifold with fibered boundary coincide if the associated family of Dirac operators on the fibers of the boundary is invertible. This answers a question raised by Piazza. Under this invertibility assumption, our method yields an index formula f…

We prove that the twisted Kahler-Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi-Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when studying the collapsing of Ricci-flat Kahler metrics on Calabi-Yau manifolds, and of th…

The paper proves Schauder estimates for Laplace-Beltrami on manifolds with fibered boundaries.

We study the Teichmüller space of negatively curved metrics on a high dimensional manifold, with applications to bundles with negatively curved fibers.

On any odd-dimensional oriented Riemannian manifold we define a volume form, which we call the odd Pfaffian, through a certain invariant polynomial with integral coefficients in the curvature tensor. We prove an intrinsic Chern-Gauss-Bonnet formula for incomplete edge singularities in terms of the odd Pfaffian on the f…

In this article, we study the higher-order regularity of the Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. We proved, using a parabolic analogue of Hein-Tosatti's work on collapsing Calabi-Yau metrics, that when the generic fibers of the Iitaka fibration are biholomorphic to each …

FiberNet integrates geometry into machine learning for clearer classification.

Study shows hypercomplex twistor spaces lack divisors and special metrics.

In this paper we prove that the Kähler-Einstein metrics for a degeneration family of Kähler manifolds with ample canonical bundles Gromov-Hausdorff converge to the complete Kähler-Einstein metric on the smooth part of the central fiber when the central fiber has only normal crossing singularities inside smooth total sp…

We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the manifold is of intermediate Kodaira dimension and has semiample canonical bundle, so…

Study asymptotics of hyperkähler geometry on singular fibers of Hitchin moduli space.

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. Results of this paper extend Whitney theorem to the case when all fibers are homeomorphic to a given compact two-dimensional manifold.

Extends differential calculus to triole algebras.

Diffusion-weighted MR imaging (DWI) is the only method we currently have to measure connections between different parts of the human brain in vivo. To elucidate the structure of these connections, algorithms for tracking bundles of axonal fibers through the subcortical white matter rely on local estimates of the fiber …

Cartan-Lie algebroids, i.e. Lie algebroids equipped with a compatible connection, permit the definition of an adjoint representation, on the fiber as well as on the tangent of the base. We call (positive) quadratic Lie algebroids, Cartan-Lie algebroids with ad-invariant (Riemannian) metrics on their fibers and base $κ$…

Given a projective hyperkahler manifold with a holomorphic Lagrangian fibration, we prove that hyperkahler metrics with volume of the torus fibers shrinking to zero collapse in the Gromov-Hausdorff sense (and smoothly away from the singular fibers) to a compact metric space which is a half-dimensional special Kahler ma…

We study the behavior of the Kähler-Ricci flow on some Fano bundle which is a trivial bundle on one Zariski open set. We show that if the fiber is $\mathbb{P}^{m}$ blown up at one point or some weighted projective space blown up at the orbifold point and the initial metric is in a suitable kähler class, then the fibers…

The space of positive Lagrangians in an almost Calabi-Yau manifold is an open set in the space of all Lagrangian submanifolds. A Hamiltonian isotopy class of positive Lagrangians admits a natural Riemannian metric $Υ$, which gives rise to a notion of geodesics. We study geodesics of positive $O_n(\mathbb{R})$ invariant…

In 1995 D. Joyce explicitly constructed a series of self-dual metrics with torus action on the connected sums of complex projective planes. In this paper we explicitly construct the twistor spaces of some of Joyce's self-dual metrics. Starting from a fiber space whose fibers are compact singular toric surfaces, we appl…

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. An extension of the Whitney's theorem to the case when all fibers are homeomorphic to some fixed compact two-dimensional manifold was proved by the…

Geometric approach to quantum thermodynamics models state spaces and processes.

In this short note, we show that given a special Kähler-Einstein degeneration with bounded geometry, for any noncentral fiber, there exists a Kähler-Ricci flow which converges to the Kähler-Einstein metric of the central fiber. As an example, Tian's deformations \cite{Tian97} of the Mukai 3-fold admit Kähler-Ricci flow…

Non-Archimedean balanced metrics approximate cscK metrics for totally degenerate abelian varieties

We study the Ricci iteration for homogeneous metrics on spheres and complex projective spaces. Such metrics can be described in terms of modifying the canonical metric on the fibers of a Hopf fibration. When the fibers of the Hopf fibration are circles or spheres of dimension 2 or 7, we observe that the Ricci iteration…