arXiv research

New class of singular complex manifolds studied with degenerate theory.

Study small eigenvalues on Kähler manifolds degenerating with induced metrics.

New method constructs degenerate Sasakian manifolds from hyperkähler bundles.

Solves a specific Dirichlet problem on Riemannian manifolds.

Study on Calabi-Yau metrics and their degenerations.

Study of symplectic manifolds degenerating into singular spaces.

Compactness theory for biharmonic maps on degenerating Einstein manifolds.

This is a short expository note about Calabi-Yau manifolds and degenerations of their Ricci-flat metrics.

We investigate degenerate special-Hermitian metrics on compact complex manifolds, in particular, degenerate Kähler and locally conformally Kähler metrics on special classes of non-Kähler manifolds.

The abstract discusses transformations on statistical and semi-Weyl manifolds with torsion.

A diffeomorphism of pseudo-Riemannian manifolds is called sectional curvature preserving if it preserves the sectional curvature of all the nondegenerate 2-planes. We consider a similar condition for degenerate 2-planes and we prove that the diffeomorphism is conformal (when the condition is fulfilled for weakly degene…

Degenerate twistor deformations of Kähler manifolds are also Kähler.

The paper explores unique properties of Kähler manifolds without shared CR-submanifolds.

This paper is the first arising from our project announced in math.AG/0211094, "Affine manifolds, log structures, and mirror symmetry." We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certai…

Uniform estimates for Calabi-Yau degenerations proved.

The paper studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.

We prove a regularity result for the Monge--Ampère equations on compact Kaehler manifolds with degenerate rhs member.

Study of polygon degeneration to segments in complex space.

Let $M$ be a hyperkaehler manifold, and $η$ a closed, positive (1,1)-form which is degenerate everywhere on $M$. We associate to $η$ a family of complex structures on $M$, called a degenerate twistor family, and parametrized by a complex line. When $η$ is a pullback of a Kaehler form under a Lagrangian fibration $L$, a…

The study shows that nonpositively curved 4-manifolds with zero Euler characteristic have degenerating Ricci curvature.

We study horospheres in hyperbolic 3-manifolds $M$ all whose ends are degenerate. Towards this, we study which almost minimizing geodesics in $M$ go through arbitrarily thin parts.

In this paper, we study the convergence of Calabi-Yau manifolds under Kähler degeneration to orbifold singularities and complex degeneration to canonical singularities (including the conifold singularities), and the collapsing of a family of Calabi-Yau manifolds.

We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.

This is a survey article of the recent progresses on the metric behaviour of Ricci-flat Kähler-Einstein metrics along degenerations of Calabi-Yau manifolds.

Two Calabi-Yau theorems for Kähler manifold degenerations.

Maximal regularity for nonuniformly parabolic problems with normal degeneration.

We shall use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge-Ampère type equations on compact Kähler manifolds.

The paper proves rigidity for warped product spaces with degenerate ends.

The study resolves a conjecture about harmonic forms on compact manifolds.

In this paper we prove that the Kähler-Einstein metrics for a toroidal canonical 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 base locus of the degeneration family is empty. We also …

Smooth functions on manifolds with degenerate singular submanifolds

Study on collapsing Calabi-Yau manifolds and their metrics.

In this work, we study Monge-Ampere equations over closed Kähler manifolds with degenerated cohomology classes. Classic results and arguments in pluripotential theory are generalized a little bit to be applied to our situation.

We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a co…

Study of J-flow on Kähler manifolds confirms energy properness.

This study solves a Dirichlet problem for specific elliptic equations on Riemannian manifolds with concave boundaries.

We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate Kähler condition are even, and the even Betti numbers are strictly positive.

We study Yamabe metrics, and the moduli space of Yamabe metrics, on an arbitrary closed 3-manifold M. The main focus is on the boundary behavior of the moduli space, i.e. the behavior of degenerating sequences of unit volume Yamabe metrics on M. It is proved that such degenerations, when non-trivial in a certain sense,…

Solves complex Monge-Ampère equations on Kähler manifolds.

Introduces a new geometric structure for statistical manifolds with degenerate metrics.

We give complete classification of C^2-regular and non-degenerate projectively Anosov flows on three dimensional manifolds. More precisely, we prove that such a flow on a connected manifold must be either an Anosov flow or represented as a finite union of $T^2 \times [0,1]$-models.

Proves a conjecture for Calabi-Yau manifolds.

In this paper we show that the convergence of complete Kahler-Einstein hypersurfaces in complex torus in the sense of Cheeger-Gromov will canonically degenerate the underlying manifolds into "pair of pants" decomposition. We also construct minimal Lagrangian tori that represent the vanishing cycles of the degeneration.

This is a survey of our recent work on degenerations of Ricci-flat Kahler metrics on compact Calabi-Yau manifolds with Kahler classes approaching the boundary of the Kahler cone.

In this Thesis, I investigate how Fano manifolds equipped with a Kahler-Einstein metric can degenerate as metric spaces (in the Gromov-Hausdorff topology) and some of the relations of this question with Algebraic Geometry, in particular in the direction of the study of moduli spaces and their compactifications.

Extends arguments to limit structure in Calabi-Yau degenerations.

This paper is a sequel to arXiv:1012.2940. We further investigate the Gromov-Hausdorff convergence of Ricci-flat Kähler metrics under degenerations of Calabi-Yau manifolds. We extend Theorem 1.1 in arXiv:1012.2940 by removing the condition on existence of crepant resolutions for Calabi-Yau varieties.

This paper analyses the convergence and degeneration of sequences of metrics on a 3-manifold, and relations of such with Thurston's geometrization conjecture. The sequences are minimizing sequences for a certain (optimal) scalar-curvature type functional and their degeneration is related to the sphere and torus decompo…