Ricci-flat Research Articles

Hitting estimates on Einstein manifolds and applications

Abstract We generalize the Benjamini–Pemantle–Peres estimate relating hitting probability and Martin capacity to the setting of manifolds with Ricci curvature bounded below. As applications we obtain: (1) a sharp estimate for the probability that Brownian motion comes close to the high curvature part of a Ricci-flat manifold, (2) a proof of an unpublished theorem of Naber that every noncollapsed limit of Ricci-flat manifolds is a weak solution of the Einstein equations, (3) an effective intersection estimate for two independent Brownian motions on manifolds with nonnegative Ricci curvature and positive asymptotic volume ratio. We also obtain generalizations of (1) and (2) for the manifolds with two-sided Ricci bounds and Einstein manifolds with nonzero Einstein constant.

Numerical metrics for complete intersection and Kreuzer–Skarke Calabi–Yau manifolds

We introduce neural networks (NNs) to compute numerical Ricci-flat Calabi–Yau (CY) metrics for complete intersection and Kreuzer–Skarke (KS) CY manifolds at any point in Kähler and complex structure moduli space, and introduce the package cymetric which provides computation realizations of these techniques. In particular, we develop and computationally realize methods for point-sampling on these manifolds. The training for the NNs is carried out subject to a custom loss function. The Kähler class is fixed by adding to the loss a component which enforces the slopes of certain line bundles to match with topological computations. Our methods are applied to various manifolds, including the quintic manifold, the bi-cubic manifold and a KS manifold with Picard number two. We show that volumes and line bundle slopes can be reliably computed from the resulting Ricci-flat metrics. We also apply our results to compute an approximate Hermitian–Yang–Mills connection on a specific line bundle on the bi-cubic.

Neural network approximations for Calabi-Yau metrics

Ricci flat metrics for Calabi-Yau threefolds are not known analytically. In this work, we employ techniques from machine learning to deduce numerical flat metrics for K3, the Fermat quintic, and the Dwork quintic. This investigation employs a simple, modular neural network architecture that is capable of approximating Ricci flat Kähler metrics for Calabi-Yau manifolds of dimensions two and three. We show that measures that assess the Ricci flatness and consistency of the metric decrease after training. This improvement is corroborated by the performance of the trained network on an independent validation set. Finally, we demonstrate the consistency of the learnt metric by showing that it is invariant under the discrete symmetries it is expected to possess.

On Douglas Warped Product Metrics

We study the new warped metric proposed by P. Marcal and Z. Shen. We obtain the differential equation of such metrics with vanishing Douglas curvature. By solving this equation, we obtain all Douglas warped product metrics. We show that Landsberg and Berwald warped product metrics are equivalent. We classify Douglas Ricci-flat metrics. Examples are included.

Bismut Ricci flat manifolds with symmetries

We construct examples of compact homogeneous Riemannian manifolds admitting an invariant Bismut connection that is Ricci flat and non-flat, proving in this way that the generalized Alekseevsky–Kimelfeld theorem does not hold. The classification of compact homogeneous Bismut Ricci flat spaces in dimension$5$is also provided. Moreover, we investigate compact homogeneous spaces with non-trivial third Betti number, and we point out other possible ways to construct Bismut Ricci flat manifolds. Finally, since Bismut Ricci flat connections correspond to fixed points of the generalized Ricci flow, we discuss the stability of some of our examples under the flow.

Levi-Civita Ricci-Flat Doubly Warped Product Hermitian Manifolds

Let M 1 , g and M 2 , h be two Hermitian manifolds. The doubly warped product (abbreviated as DWP) Hermitian manifold of M 1 , g and M 2 , h is the product manifold M 1 × M 2 endowed with the warped product Hermitian metric G = f 2 2 g + f 1 2 h , where f 1 and f 2 are positive smooth functions on M 1 and M 2 , respectively. In this paper, the formulae of Levi-Civita connection, Levi-Civita curvature, the first Levi-Civita Ricci curvature, and Levi-Civita scalar curvature of the DWP-Hermitian manifold are derived in terms of the corresponding objects of its components. We also prove that if the warped function f 1 and f 2 are holomorphic, then the DWP-Hermitian manifold is Levi-Civita Ricci-flat if and only if M 1 , g and M 2 , h are Levi-Civita Ricci-flat manifolds. Thus, we give an effective way to construct Levi-Civita Ricci-flat DWP-Hermitian manifold.

Families of singular Kähler–Einstein metrics

Refining Yau’s and Kołodziej’s techniques, we establish very precise uniform a priori estimates for degenerate complex Monge–Ampère equations on compact Kähler manifolds, that allow us to control the blow up of the solutions as the cohomology class and the complex structure both vary. We apply these estimates to the study of various families of possibly singular Kähler varieties endowed with twisted Kähler–Einstein metrics, by analyzing the behavior of canonical densities, establishing uniform integrability properties, and developing the first steps of a pluripotential theory in families. This provides interesting information on the moduli space of stable varieties, extending works by Berman–Guenancia and Song, as well as on the behavior of singular Ricci-flat metrics on (log) Calabi–Yau varieties, generalizing works by Rong–Ruan–Zhang, Gross–Tosatti–Zhang, Collins–Tosatti and Tosatti–Weinkove–Yang.

COHOMOGENEITY ONE KÄHLER-RICCI SOLITONS UNDER A HEISENBERG GROUP ACTION AND RELATED METRICS

We find complete expanding Kähler-Ricci solitons of two types. The first are of cohomogeneity one under the action of the (2m - 1)-dimensional Heisenberg group or a certain quotient of it, for any m ≥ 2. The second type, which generalizes the first, reside on ℂ*-bundles over compact Kähler Ricci-flat manifolds, admit an S1 action by isometries, and have two ends. Both types have the local structure stemming from an ansatz first described in [20]. We give curvature bounds for the first type, and asymptotics for both types.

Calabi–Yau metrics, energy functionals and machine learning

In this paper, we apply machine learning to the problem of finding numerical Calabi–Yau metrics. We extend previous work on learning approximate Ricci-flat metrics calculated using Donaldson’s algorithm to the much more accurate “optimal” metrics of Headrick and Nassar. We show that machine learning is able to predict the Kähler potential of a Calabi–Yau metric having seen only a small sample of training data.

Local curvature estimates for the Ricci-harmonic flow

In this paper we give an explicit bound of Δg(t)u(t) and the local curvature estimates for the Ricci-harmonic flow ∂tg(t)=−2Ricg(t)+4du(t)⊗du(t),∂tu(t)=Δg(t)u(t)under the condition that the Ricci curvature is bounded along the flow. In the second part these local curvature estimates are extended to a class of generalized Ricci flow, introduced by the author Li (2019), whose stable points give Ricci-flat metrics on a complete manifold, and which is very close to the (K,N)-super Ricci flow recently defined by Li and Li (0000). Next we propose a conjecture for Einstein’s scalar field equations motivated by a result in the first part and the bounded L2-curvature conjecture recently solved by Klainerman et al. (2015). In the last part of this paper, we discuss the forward and backward uniqueness for the Ricci-harmonic flow.

Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups

AbstractLetXbe a compact Kähler space with klt singularities and vanishing first Chern class. We prove the Bochner principle for holomorphic tensors on the smooth locus ofX: any such tensor is parallel with respect to the singular Ricci-flat metrics. As a consequence, after a finite quasi-étale coverXsplits off a complex torus of the maximum possible dimension. We then proceed to decompose the tangent sheaf ofXaccording to its holonomy representation. In particular, we classify thoseXwhich have strongly stable tangent sheaf: up to quasi-étale covers, these are either irreducible Calabi–Yau or irreducible holomorphic symplectic. As an application of these results, we show that ifXhas dimension four, then it satisfies Campana’s Abelianity Conjecture.

Deformations of Q-curvature II

This is the second article of a sequence of research on deformations of Q-curvature. In the previous one, we studied local stability and rigidity phenomena of Q-curvature. In this article, we mainly investigate the volume comparison with respect to Q-curvature. In particular, we show that volume comparison theorem holds for metrics close to strictly stable positive Einstein metrics. This result shows that Q-curvature can still control the volume of manifolds under certain conditions, which provides a fundamental geometric characterization of Q-curvature. Applying the same technique, we derive the local rigidity of strictly stable Ricci-flat manifolds with respect to Q-curvature, which shows the non-existence of metrics with positive Q-curvature near the reference metric.

Calabi-Yau CFTs and random matrices

Using numerical methods for finding Ricci-flat metrics, we explore the spectrum of local operators in two-dimensional conformal field theories defined by sigma models on Calabi-Yau targets at large volume. Focusing on the examples of K3 and the quintic, we show that the spectrum, averaged over a region in complex structure moduli space, possesses the same statistical properties as the Gaussian orthogonal ensemble of random matrix theory.

Hermitian\u2013Yang\u2013Mills Connections on Collapsing Elliptically Fibered K3 Surfaces

Let Xrightarrow {{mathbb {P}}}^1 be an elliptically fibered K3 surface, admitting a sequence omega _{i} of Ricci-flat metrics collapsing the fibers. Let V be a holomorphic SU(n) bundle over X, stable with respect to omega _i. Given the corresponding sequence Xi _i of Hermitian–Yang–Mills connections on V, we prove that, if E is a generic fiber, the restricted sequence Xi _i|_{E} converges to a flat connection A_0. Furthermore, if the restriction V|_E is of the form oplus _{j=1}^n{mathcal {O}}_E(q_j-0) for n distinct points q_jin E, then these points uniquely determine A_0.

Extending $$\Lambda (t)-$$CDM to the inflationary epoch using dynamical foliations and a pre-inflationary vacuum energy from 5D geometrical vacuum as a unifying mechanism

In this paper assuming a 5D quantum pre-inflationary vacuum energy, we propose a manner to extend some \(\Lambda (t)\)-CDM models to the inflationary period by using dynamical foliations of the five-dimensional (5D) Ricci-flat space-time manifold, regarding a non-compact extra space-like coordinate. In this formalism, we achieve also a geometrical unification of inflation and the present accelerating epoch. In this approach, inflation is generated by a pre-inflationary quantum vacuum energy that maintains the 5D classical vacuum on cosmological scales. We obtain from geometrical conditions that we can model the presence of the pre-inflationary vacuum energy in 4D as a dynamical cosmological constant. In this model, the 4D inflationary period is governed by a power law expansion and for certain values of some parameters of the model, we obtain an spectral index satisfying \(0.9607\le n_s\le 0.9691\) and a scalar-to-tensor ratio \(r=0.098\), values that fit well according to Planck 2018 results. The 4D inflationary potential is induced for the 5D geometry and the 4D pre-inflationary potential is determined by the model and its contribution is necessary so that \(n_s\) and r can fit the observational data. We also show that in this theoretical framework, the present acceleration in the expansion of the universe can be explained due to a remanent of this pre-inflationary vacuum energy scaled to the present epoch and that its description can be done with the same \(\Lambda (t)\). In this period, we obtain a deceleration parameter in agreement with Planck 2018 data under certain restrictions of the parameters of the model. From the geometrical point of view, \(\Lambda (t)\) is depending on the dynamical foliation of the 5D space-time manifold.

Canonical Identification at Infinity for Ricci-Flat Manifolds

We give a natural way to identify between two scales, potentially arbitrarily far apart, in a non-compact Ricci-flat manifold with Euclidean volume growth when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation.

Flat left-invariant pseudo-Riemannian metrics on quadratic Lie groups

A flat quadratic Lie algebra (g,〈,〉,k) is a Lie algebra g endowed with a flat pseudo-Euclidean metric 〈,〉 and a quadratic structure k. In geometrical terms, it is a Lie algebra of a Lie group endowed with both a flat left-invariant pseudo-Riemannian metric and a bi-invariant metric. It is known that if a Lie algebra admits a symplectic form and a quadratic structure then it is a flat quadratic Lie algebra. In this paper, we show that there exist non-abelian Lie algebras which admit a flat Lorentzian metric and a quadratic structure. If g is such Lie algebra, then we show that g is a trivial central extension of R⋊H2k+1 and [g,g]≃H2k+1, where H2k+1 is the Heisenberg Lie algebra and k≥2. In particular, we prove that g is 3-step solvable, not nilpotent and hence admits no symplectic structure. We show that there exists only one non-abelian Lie algebra of dimension n≤6 which admits a flat Lorentzian metric and a quadratic structure. Using a variant of the double extension method, which conserves both the flat and the quadratic metrics, we prove that any Lie algebra which admits a flat Lorentzian metric and a quadratic structure, is a double extension of an abelian Lie algebra and we construct a large class of such (non-abelian) Lie algebras in higher dimensions. Oscillator Lie algebras denoted by gλ constitute an important class of quadratic Lie algebras which are of the form R⋊H2k+1. It is known that gλ admits no flat Lorentzian metric. We show in this paper that gλ admits a Ricci-flat Lorentzian metric. The paper also study the class of quadratic 2-step nilpotent Lie algebras (g,k) which constitute examples of flat quadratic Lie algebras since k is also flat in this case.

Complete noncompact G2-manifolds from asymptotically conical Calabi–Yau 3-folds

We develop a powerful new analytic method to construct complete noncompact Ricci-flat 7-manifolds, more specifically G2-manifolds, that is, Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction gives the first general analytic construction of complete noncompact Ricci-flat metrics in any odd dimension and establishes a link with the Cheeger–Fukaya–Gromov theory of collapse with bounded curvature. The construction starts with a complete noncompact asymptotically conical Calabi–Yau 3-fold B and a circle bundle M→B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics gϵ on M that collapses with bounded curvature as ϵ→0 to the original Calabi–Yau metric on the base B. The G2-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics; these are the natural higher-dimensional analogues of the asymptotically locally flat (ALF) metrics that are well known in 4-dimensional hyper-Kähler geometry. We give two illustrations of the strength of our method. First, we use it to construct infinitely many diffeomorphism types of complete noncompact simply connected G2-manifolds; previously only a handful of such diffeomorphism types was known. Second, we use it to prove the existence of continuous families of complete noncompact G2-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete noncompact G2-metrics were known.

Existence and uniqueness of asymptotically flat toric gravitational instantons

We prove uniqueness and existence theorems for four-dimensional asymptotically flat, Ricci-flat, gravitational instantons with a torus symmetry. In particular, we prove that such instantons are uniquely characterised by their rod structure, which is data that encodes the fixed point sets of the torus action. Furthermore, we establish that for every admissible rod structure there exists an instanton that is smooth up to possible conical singularities at the axes of symmetry. The proofs involve adapting the methods that are used to establish black hole uniqueness theorems, to a harmonic map formulation of Ricci-flat metrics with torus symmetry, where the target space is directly related to the metric (rather than auxiliary potentials). We also give an elementary proof of the nonexistence of asymptotically flat toric half-flat instantons. Finally, we derive a general set of identities that relate asymptotic invariants such as the mass to the rod structure.

Four-dimensional Einstein manifolds with Heisenberg symmetry

We classify Einstein metrics on mathbb {R}^4 invariant under a four-dimensional group of isometries including a principal action of the Heisenberg group. We consider metrics which are either Ricci-flat or of negative Ricci curvature. We show that all of the Ricci-flat metrics, including the simplest ones which are hyper-Kähler, are incomplete. By contrast, those of negative Ricci curvature contain precisely two complete examples: the complex hyperbolic metric and a metric of cohomogeneity one known as the one-loop deformed universal hypermultiplet.