arXiv research

Calculates characteristic classes of flat vector bundles on complex fibers.

Counterexample disproves conjecture on flat metrics and fiber bundles.

New examples show not all homology fiber bundles are topological.

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…

Study on noncompact steady quasi-Einstein manifolds with positive Ricci curvature.

Researchers create metrics collapsing to a line, detailing near-fiber behavior.

Paper finds unique maximal hypersurfaces in open spacetimes.

The study proves conditions for quasi-Einstein manifolds with specific structures to be Einstein.

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…

Generalized complex structures on certain torus bundles are explored.

Study on Hermitian-Yang-Mills connections on collapsing K3 surfaces.

Introduces symplectic flatness for connections over symplectic manifolds.

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…

Maps are shown to be Riemannian products with Ricci-flat fibers.

New invariants for families of flat connections constructed using fiber integration.

Let $X$ be a Kähler manifold which is fibered over a complex manifold $Y$ such that every fiber is a Calabi-Yau manifold. Let $ω$ be a fixed Kähler form on $X$. By Yau's theorem, there exists a unique Ricci-flat Kähler form $ρ\vert_{X_y}$ for each fiber, which is cohomologous to $ω\vert_{X_y}$. This family of Ricci-fla…

Study Higgs sections and flat sections for nonlinear harmonic bundles.

A Seifert manifold is a 3-dimensional manifold with a circle action. It is a circle bundle (with singularities) over a 2-dimensional orbifold. In this note, we discuss a generalized Seifert manifolds. By definition, they have bundle-like structures whose fibers are infra- homogeneous spaces; that is, the fibers are fla…

Study shows convergence of Yang-Mills connections on K3 surfaces under fiber collapse.

Proves a complex flat vector bundle theorem using index theory.

Constructs a new type of metric for elliptic surfaces.

In this paper we prove that any complete locally conformally flat quasi-Einstein manifold of dimension $n\geq 3$ is locally a warped product with $(n-1)$-dimensional fibers of constant curvature. This result includes also the case of locally conformally flat gradient Ricci solitons.

Study heat flow on collapsing K3 surfaces, handling conic singularities.

Extends Higgs fields theory to complex fiber bundles.

We describe pairs (p,n) such that n-dimensional affine space is fibered by pairwise skew p-dimensional affine subspaces. The problem is closely related with the theorem of Adams on vector fields on spheres and the Hurwitz-Radon theory of composition of quadratic forms.

In this paper we show that a compact warped product Einstein manifold with vanishing Bach tensor of dimension $n \geq 4$ is a finite quotient of a warped product with $(n-1)$-dimensional Einstein fiber. The fiber has constant curvature if $n=4$.

We characterize compact locally conformal parallel $G_2$ (respectively, $Spin(7)$) manifolds as fiber bundles over $S^1$ with compact nearly Kähler (respectively, compact nearly parallel $G_2$) fiber. A more specific characterization is provided when the local parallel structures are flat.

This paper extends curvature results to higher fiber dimensions.

The study characterizes and classifies specific types of manifolds using conformal and quasi-Einstein properties.

We give an integral representaion of the zeta-reguralized determinant of Laplacians on three dimensional Heisenberg manifolds, and study a behaivior of the values when we deform the uniform discrete subgroups. Heiseberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typ…

Study connects $G_2$-structures to flat connections on compact 3-manifolds.

Study maximal hypersurfaces in open spacetimes using a maximum principle.

The paper studies critical points and flows of a $G_2$-Hilbert functional on manifolds with circle actions.

Let G be an n-dimensional crystallographic group (n-space group). If G is a Z-reducible, then the flat n-orbifold E^n/G has a nontrivial fibered orbifold structure. We prove that this structure can be described by a generalized Calabi construction, that is, E^n/G is represented as the quotient of the Cartesian product …

Manifolds with fibered cusps are a class of complete noncompact Riemannian manifolds including all locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asympt…

We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the re…

Develops method to create non-Abelian Ricci-flat graphs via bundles.

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

The paper shows Euler classes for homeomorphisms of Seifert fibered 3-manifolds are unbounded.

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…

Study of twisted Kähler-Einstein metrics on Calabi-Yau spaces with singularities.

Paper defines conditions for warped product gradient Yamabe solitons.

We classify homotopes of classical symmetric spaces (studied in Part I of this work). Our classification uses the fibered structure of homotopes: they are fibered as symmetric spaces, with flat fibers, over a non-degenerate base; the base spaces correspond to inner ideals in Jordan pairs. Using that inner ideals in cla…

Study on K3 surfaces with collapsing metrics and nilpotent structures.

The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution.

Given a polarized family of varieties over the unit disc, smooth except over the origin and with smooth fibers Calabi-Yau, we show that the origin lies at finite Weil-Petersson distance if and only if after a finite base change the family is birational to one with central fiber a Calabi-Yau variety with at worst canoni…

Electrostatic systems with specific tensors are locally conformally flat.

Local Sasaki-Ricci solitons are $η$-Einstein in certain fiber products of homogeneous Sasakian manifolds.