arXiv research

New metrics found on non-Kähler Calabi-Yau manifolds.

New methods find Ricci-flat metrics on specific Lie groups.

Study stability and instability of Ricci-flat metrics under generalized Ricci flow.

The paper constructs flat metrics on orbifolds and resolutions.

Simply connected moduli space of Ricci flat metrics on K3 surfaces.

Study on Calabi-Yau metrics and their degenerations.

Study on how incomplete smooth metrics degenerate from asymptotically conical Ricci-flat Kähler metrics.

We propose two conjectures about Ricci-flat metrics: Conjecture 1: A Ricci-flat projectively induced metric is flat. Conjecture 2: A Ricci-flat metric on an $n$-dimensional complex manifold such that the $a_{n+1}$ coefficient of the TYZ expansion vanishes is flat. We verify Conjecture 1 (see Theorem 1.1) under the assu…

In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Kahler Calabi-Yau surfaces or Hopf surfaces.

Study K3 surfaces and their metrics, focusing on dynamics.

Study on stability of ALE Ricci-flat metrics using a modified Perelman's λ-functional.

We prove that a crepant resolution of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat Kähler metrics on crepant resolutions of X=C^n /G, where G is a finite subgroup…

We determine all Ricci flat left invariant Lorentzian metrics on simply connected 2-step nilpotent Lie groups. We show that the $2k+1$-dimensional Heisenberg Lie group $H_{2k+1}$ carries a Ricci flat left invariant Lorentzian metric if and only if $k=1$. We show also that for any $2\leq q\leq k$, $H_{2k+1}$ carries a R…

Constructs Einstein metrics on manifolds with specific orbits.

The paper constructs a new Ricci-flat metric on almost abelian Lie groups.

Proves orbifold singularities for Ricci-flat metrics on certain Kähler varieties.

The paper proves volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.

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…

In this note, we obtain existence results for complete Ricci-flat Kahler metrics on crepant resolutions of singularities of Calabi-Yau varieties. Furthermore, for certain asymptotically flat Calabi-Yau varieties, we show that the Ricci-flat metric on the resolved manifold has the same asymptotic behavior as the initial…

Using methods of A. Grigor'yan and L. Saloff-Coste we prove that on a manifold with a conical end the heat kernel has a Gaussian bound. This result is applied to asymptotically conical Kähler manifolds. It is a result of the author and R. Goto that a crepant resolution of a Ricci-flat Kähler cone admits a Ricci-flat Kä…

Classifies Einstein metrics on R^4 with Heisenberg symmetry, finding incomplete Ricci-flat metrics and two complete negative-curvature examples.

Neural nets approximate Ricci flat metrics for Calabi-Yau manifolds.

The study computes indicial roots and metric convergence orders for Ricci-flat conifolds.

Let N_0 = C^2/H be an isolated quotient singularity with H in U (2) a finite subgroup. We show that for any Q-Gorenstein smoothings of N_0 a nearby fiber admits ALE Ricci-flat Kahler metrics in any Kahler class. Moreover, we generalize Kronheimer's results on hyperkahler 4-manifolds, by giving an explicit classificatio…

Method finds approximate Ricci-flat metrics on Calabi-Yau manifolds.

Study constructs non-Riemannian Finsler metrics using warped product.

New functional proves mass positivity for ALE metrics.

This work shows all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces are known families.

We study a class of asymptotically cylindrical Ricci-flat Kähler metrics arising on quasiprojective manifolds. Using the Calabi--Yau geometry and analysis and the Kodaira--Kuranishi--Spencer theory and building up on results of N.Koiso for the case of compact manifolds, we show that under rather general hypotheses any …

New metrics with non-negative scalar curvature are always Ricci-flat on certain surgeries.

We define a system of ODE that gives Einstein 4-dimensional metrics. We found new Ricci-flat incomplete metric of cohomogeneity 1 in explicit formulas and study its characteristics.

We study the behaviour of families of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold when the Kahler classes degenerate to the boundary of the ample cone. We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ri…

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli-Chern class on compact complex manifolds, and proved that the $(1,1)$ curvature form of the Levi-Civita connection represents the first Aeppli-Chern class which is a natural link between Riemannian geometry and complex geometry. …

We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over $\mathbb{CP}^2$, $\mathbb{HP}^2$ and $\mathbb{OP}^2$ with respective principal orbits $G/K$ the Wallach spaces $SU(3)/T^2$, $Sp(3)/(Sp(1)Sp(1)Sp(1))$ and $F_4/\mathrm{spin}(8)$. Almost all the …

Researchers found a non-Ricci-flat Einstein metric on a 7D nilpotent Lie group.

Bismut Einstein metrics on complex manifolds are Kähler Einstein or Bismut Ricci flat.

We prove the existence of a (unique) S^1-invariant Ricci-flat Kaehler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kaehler manifold X, extending the metric on X.

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.

We give a description of all $G$-invariant Ricci-flat Kähler metrics on the canonical complexification of any compact Riemannian symmetric space $G/K$ of arbitrary rank, by using some special local $(1,0)$ vector fields on $T(G/K)$. As the simplest application, we obtain the explicit description of the set of all compl…

In this article, we introduce a new method (based on Perelman's lambda-functional) to study the stability of compact Ricci-flat metrics. Under the assumption that all infinitesimal Ricci-flat deformations are integrable we prove: (A) a Ricci-flat metric is a local maximizer of lambda in a C^2,alpha-sense iff its Lichne…

Blowing up flat metrics yields balanced ones with constant curvature.

Study finds 132 complex invariant Einstein metrics on a specific flag manifold and constructs Ricci-flat metrics.

The author has proved that a crepant resolution Y of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H^2_c(Y,\R). These manifolds are generalizations of the Ricci-flat ALE Kähler spaces known by the work of P. Kronheimer, D. Joyce and others. …

Study on collapsing Calabi-Yau manifolds and their metrics.

In this short article, we improve the dynamical stability and instability results for Ricci-flat metrics under Ricci flow proved by Sesum and Haslhofer, getting rid of the integrability assumption.

We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $\leq6$, every nice nilpotent Lie group of dimension $\leq7$ and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct inf…

In this paper, we study the behavior of Ricci-flat Kähler metrics on Calabi-Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa's conjecture: Ricci-flat Calabi-Yau manifolds related by extremal transitions and flops can be connected by a path c…

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…