Igor Belegradek, Guofang Wei
We construct metrics of positive Ricci curvature on some vector bundles over tori (or more generally, over nilmanifolds). This gives rise to the first examples of manifolds with positive Ricci curvature which are homotopy equivalent but not homeomorphic to manifolds of nonnegative sectional curvature.
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We completely describe the signatures of the Ricci curvature of left-invariant Riemannian metrics on arbitrary real nilpotent Lie groups. The main idea in the proof is to exploit a link between the kernel of the Ricci endomorphism and closed orbits in a certain representation of the general linear group, which we prove using the `real GIT' framework for the Ricci curvature of nilmanifolds.
In this note we prove that any left-invariant almost Hermitian structure on a 2-step nilmanifold is Ricci-flat with respect to the Chern connection and that it is Ricci-flat with respect to another canonical connection if and only if it is cosymplectic.
In this note we observe that on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure the (1,1)-part of the Bismut-Ricci form is seminegative definite. As application we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long time solution to the pluriclosed flow in 2-step nilmanifolds.
Randers Ricci soliton homogeneous nilmanifolds
本文研究左不变 Randers 度量下的幂零李群，证明在 Ricci 流方程唯一解时，Ricci 孤子等价于半代数 Ricci 孤子。
We prove that the Ricci flow g(t) starting at any metric on the euclidean space that is invariant by a transitive nilpotent Lie group N, can be obtained by solving an ODE for a curve of nilpotent Lie brackets. By using that this ODE is the negative gradient flow of a homogeneous polynomial, we obtain that g(t) is type-III, and, up to pull-back by time-dependent diffeomorphisms, that g(t) converges to the flat metric, and the rescaling |R(g(t))|g(t) converges smoothly to a Ricci soliton, uniformly on compact sets. The Ricci soliton limit is also invariant by some transitive nilpotent Lie group,…
This is partly an expository paper, where the authors' work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-flat metrics on nilpotent Lie groups of dimension $\leq8$ are obtained. Some related open questions are presented.
We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product and the evolution of structure constants, as well as the evolution of these quantities modulo rescaling. We set up systems of O.D.E.'s for some of these flows and describe their qualitative properties. We also present some explicit solutions for the evolution of soliton metrics under the Ricci flow.
We answer in the affirmative the question posed by Conti and Rossi on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric $g$ of signature $(3, 4)$ on a nilpotent Lie group of dimension 7, such that $g$ is Einstein and not Ricci-flat. We show that the pseudo-metric $g$ cannot be induced by any left-invariant closed $G_2^*$-structure on the Lie group. Moreover, some results on closed and harmonic $G_2^*$-structures on an arbitrary 7-manifold $M$ are given. In particular,…
The heterotic G$_2$-system on 2-step nilmanifolds endowed with principal torus bundles
研究赋予主环面丛的 7 维 2 步幂零流形上的杂化 G₂ 系统，证明特定条件下不变 G₂ 结构余闭，讨论 7 维 2 步幂零李代数解的存在性并给出含常 dilaton 函数的例子。
A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of $G$-invariant complex structures which satisfy quaternionic relations). We prove that a hypercomplex nilmanifold admits an HKT (hyperkahler with torsion) metric if and only if the underlying hypercomplex structure is abelian. Moreover, any $G$-invariant HKT-metric on a nilmanifol…
Let (N,J) be a real 2n-dimensional nilpotent Lie group endowed with an invariant complex structure. A left-invariant Riemannian metric on N compatible with J is said to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics on (N,J) with the same scalar curvature. In this paper, we determine all complex structures that admit a minimal compatible metric on 6-dimensional nilpotent Lie groups.
We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fibre bundles over complex manifolds. We apply our results in the case of left-invariant generalized complex structures on nilmanifolds and to their space of small deformations.
Starting from an even definite lattice, we construct a principal circle bundle covered by a certain three-step nilpotent Lie group G. On the base space, which is again a nilmanifold, we then study the Dirac operator twisted by the associated complex line bundles. Noting that the whole situation fibers over the circle, we are able to determine the reduced eta-invariant of these Dirac operators in the adiabatic limit. As an application, we consider the total space of the circle bundle, equipped with a parallelism induced by G, as an element in the stable homotopy groups of the sphere and use the…
We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to $\pm\infty$ or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities. Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Lap…
Sectional Curvature Pinching of Two-Step Nilmanifolds
本文研究二步幂零流形的截面曲率拼挤常数，证明其位于[-3,-3/2]，上界具有刚性，下界非刚性，并给出代数刻画。
Let (N,g) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their `almost' versions). We define a left invariant Riemannian metric on N compatible with g to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the `invariant Ricci' flow and are characterized as the critical points of a natural variational problem. The uniqueness…
In this paper we compute the Ricci flow formulas for invariant metrics on prinicpal $G$-bundles compatible with the connection. Our primary focus is on torus bundles which we use to study a notion of Bakry-Émery Ricci flow as well as Ricci flow on circle bundles over Kähler-Einstein manifolds. The latter application gives us solutions to Ricci flows on Heisenberg groups and implicit solutions to Ricci flows on Berger 3-spheres and several other 3-dimensional manifolds.
For Fano homogeneous toric bundles, we obtain a formula of the greatest lower bound on Ricci curvature. We also give a criteria for the ampleness of a kind of line bundles over general homogeneous toric bundles.
Quasi-projective nilmanifolds
本文研究了准投影品种M与非紧nilmanifold的同胚关系，证明在特定条件下M可与平凡纤维丛同胚，并确定了相关李群的维度限制。
We construct new examples of manifolds of positive Ricci curvature which, topologically, are vector bundles over compact manifolds of almost nonnegative Ricci curvature. In particular, we prove that if E is the total space of a vector bundle over a compact manifold of nonnegative Ricci curvature, then the product of E and R^p admits a complete metric of positive Ricci curvature for all large p.
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.
In this paper, we propose a method of studying the modified Kahler-Ricci flow on projective bundles and give the explicit equation from the view point of symplectic geometry.
We exhibit families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds.
The Ricci flow for circle bundles over surfaces
本文研究曲面上圆周丛的归一化里奇流方程，求解该方程并分析解的渐近行为及其与模型几何的联系。
It is proved that an homogeneous toric bundles over a flag manifold G^\C/P admits a Kaehler-Ricci solitonic metric if and only if it is Fano. In particular, an homogeneous toric bundle of this kind is Kaehler-Einstein if and only if it is Fano and its Futaki invariant vanishes identically.
Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. In [L1], J. Lauret proved that minimal metrics (if any) are unique up to isometry and scaling. This uniqueness allows us to distinguish two complex structures with Riemannian data, giving rise to a great deal of invariants. We show how to use a Riemannian invariant: t…
We show the existence and uniqueness of a one-parameter family of smooth complete $U(1)$-invariant gradient steady Ricci solitons on the total space of any complex line bundle over a Fano Kähler-Einstein base with first Chern class proportional to that of the base. These solitons are non-Kähler except on the total space of the canonical bundle.
The aim of this paper is to present a direct and simple proof of a result concerning the existence of metrics of positive Ricci curvature on the total space of fiber bundles with compact structure groups. In particular, it also generalizes and puts in a unified framework the results in Nash \cite{nash} and Poor \cite{poor}. With the intention of disseminating this result, we apply it to build new examples of manifolds with positive Ricci curvature, including bundles whose base consists of gradient shrinking Ricci solitons.
We study the behavior of the Kähler-Ricci flow on compact manifolds developing finite-time singularities, in particular, when the flow contracts exceptional divisors or collapses the Fano fibers of a holomorphic fiber bundle. We present a technique using holomorphic vector fields to prove estimates related to the work of Song-Weinkove and Fu-Zhang.
We study the behaviour of the Kähler-Ricci flow on projective bundles. We show that if the initial metric is in a suitable Kähler class, then the fibers collapse in finite time and the metrics converge subsequentially in the Gromov-Hausdorff sense to a metric on the base.
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 infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie groups ${\rm SL}(n)$, ${\rm SO}(p,q)$, ${\rm Sp}(n,\mathbb R)$. Most of these metrics are shown not to be flat.
Considering pseudo-Riemannian $g$-natural metrics on tangent bundles, we prove that the condition of being Ricci soliton is hereditary in the sense that a Ricci soliton structure on the tangent bundle gives rise to a Ricci soliton structure on the base manifold. Restricting ourselves to some class of pseudo-Riemannian $g$-natural metrics, we show that the tangent bundle is a Ricci soliton if and only if the base manifold is flat and the potential vector field is a complete lift of a conformal vector field. We give then a classification of conformal vector fields on a flat Riemannian manifold. …
We show that all the small covers which are infra-nilmanifolds are exactly real Bott manifolds. This implies that any small cover which admits a flat Riemannian metric must be a real Bott manifold. In addition, we will study small covers which admit Riemannian metrics with positive or nonnegative Ricci curvature or sectional curvature. We will see that these geometric conditions put very strong restrictions on the topology of the small covers and the combinatorial structure of the underlying simple polytopes. Similar geometric problems are also studied for the real moment-angle complex of an a…
We give an explicit description of all complete $G$-invariant Ricci-flat Kähler metrics on the tangent bundle $T(G/K)\cong G^\bbC/K^\bbC$ of rank-one Riemannian symmetric spaces $G/K$ of compact type, in terms of associated vector-functions.
Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature
本文通过构造新泛函研究幂零主丛上里奇流 blowdown 极限的刚性，证明三维海森堡群情形下极限为扩张里奇孤子，并结合 Lott 的阿贝尔结果完成四维情形的完整局部分类。
Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and diameter, and almost non-negative scalar curvature. Let r=1 if n=2,3 and r=2^{[n/2]-1}+1 if n\geq 4. We show that if the square of the Dirac operator on such a manifold has $r$ small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a non-trivial spin structure, then there exists a uniform lower bound on the r-th eigenvalue of the square of the Dirac operator. If a manifold with a…
In this paper we modify the coordinate construction in our previous paper on the universal moduli space of pair consisting of a Riemann Surfaces and a stable holomorphic bundles on the Riemann Surface, so as to produce a new set of coordinates, which are in fact K\" ahler coordinates on this universal moduli space. Further, we give a functional determinant formula for the Ricci potential of the universal moduli space.
Positive Intermediate Ricci Curvature on Fibre Bundles
本文证明了浸没度量的典型变分型结果，结合手术技术为高连通流形和 exotic 球面等例子建立了正中间里奇曲率度量的存在性，并将正里奇曲率模空间结果推广至正中间情形。
We generalize the circle bundle examples of ancient solutions of the Ricci flow discovered by Bakas, Kong, and Ni to a class of principal torus bundles over an arbitrary finite product of Fano Kähler-Einstein manifolds studied by Wang and Ziller in the context of Einstein geometry. As a result, continuous families of $κ$-collapsed and $κ$-noncollapsed ancient solutions of type I are obtained on circle bundles for all odd dimensions $\geq 7$. In dimension $7$ such examples moreover exist on pairs of homeomorphic but not diffeomorphic manifolds. Continuous families of $κ$-collapsed ancient solut…
Infinitely many non-collapsed steady Ricci solitons on complex line bundles
研究在复射影空间上的复线丛O(k)上构造无限多个非收缩的里奇溶剂，并证明其具有渐近锥形的里奇平坦度量。
We produce new non-Kähler complete steady gradient Ricci solitons whose asymptotics combine those of the Bryant solitons and the Hamilton cigar. We also obtain a family of complete Ricci-flat metrics with asymptotically locally conical asymptotics. Finally, we obtain numerical evidence for complete steady soliton structures on the vector bundles whose distance sphere bundles are respectively the twistor and ${\rm Sp}(1)$ bundles over quaternionic projective space.
Expanding Ricci solitons and Higgs bundles
本文研究了具有幂零对称性的扩展里齐流形与Higgs丛的关系，通过建立对应关系产生新的非局部同构例子，并在四维情况下给出完整描述。
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 complete $\mathrm{SO}(3)$-invariant Ricci-flat Kähler metrics on $T{\mathbb S}^2$, which includes the well-known Eguchi-Hanson-Stenzel metrics and a new one-parameter family of metrics.
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 collapse in finite time and the metrics converge sub-sequentially in Gromov-Hausdorff sense to a metric on the base.
In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how one may determine the existence of such a metric by analyzing algebraic properties of the Lie algebra in question and infinitesimal deformations of any initial metric. Our second main result concerns the isometry groups of such distinguished metrics. Among the completely solvable unimodular Lie…
We show that the moduli space of positive Ricci curvature metrics on all the total spaces of $S^7$-bundles over $S^8$ which are rational homology spheres has infinitely many path components. Furthermore, we carry out the diffeomorphism classification of quotients of Milnor spheres by a certain involution and show that the moduli space of metrics of non-negative sectional on them has infinitely many path components. Finally, a diffeomorphism finiteness result is obtained on quotients of Shimada spheres by the same type of involution and we show that for the types that can be expressed by an inf…
This is the first of a sequence of two papers. Here, a simple algebraic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold G^C/P is provided in terms of symplectic data. The result of this paper is used in the second paper, where it is proved that an homogeneous toric bundle over a flag manifold admits a Kaehler-Ricci solitonic metric if and only if it is Fano.
Gromov-Hausdorff limit of orthonormal frame bundles of non-collapsed manifolds with bounded Ricci curvature
研究具有双边有界Ricci曲率的非坍缩流形的正交标架丛在几乎标准度量下的Gromov-Hausdorff极限空间，证明其与Ricci极限空间类似，奇异集余维数≥4且补集为开稠密的C^{1,α}黎曼流形。
We classify nilmanifolds with an invariant symplectic half-flat structure. We solve the half-flat evolution equations in one example, writing down the resulting Ricci-flat metric. We study the geometry of the orbit space of 6-manifolds with an SU(3)-structure preserved by a U(1) action, giving characterizations in the symplectic half-flat and integrable case.
A left invariant metric on a nilpotent Lie group is called minimal, if it minimizes the norm of the Ricci tensor among all left invariant metrics with the same scalar curvature. Such metrics are unique up to isometry and scaling and the groups admitting a minimal metric are precisely the nilradicals of (standard) Einstein solvmanifolds. If $N$ is endowed with an invariant symplectic, complex or hypercomplex structure, then minimal compatible metrics are also unique up to isometry and scaling. The aim of this paper is to give more evidence of the existence of minimal metrics, by presenting seve…
We produce new non-Kähler, non-Einstein, complete expanding gradient Ricci solitons with conical asymptotics and underlying manifold of the form $\R^2 \times M_2 \times \cdots \times M_r$, where $r \geq 2$ and $M_i$ are arbitrary closed Einstein spaces with positive scalar curvature. We also find numerical evidence for complete expanding solitons on the vector bundles whose sphere bundles are the twistor or ${\rm Sp}(1)$ bundles over quaternionic projective space.
In this note, using Calabi's method, we construct rotationally symmetric Kahler-Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao and Feldman-Ilmanen-Knopf.
Explicit complete Ricci-flat metrics and Kähler-Ricci solitons on direct sum bundles
该研究在Kähler-Einstein Fano流形上由典范根丛幂次直和构成的向量丛全空间中，利用哈密顿2-形式理论构造出多种完备Kähler-Ricci孤立子与Calabi-Yau度量，得到了具有不同体积增长特性的新几何例子。
Finite-Time Singularities of the Kähler--Ricci Flow on Fano Bundles
该研究针对解析极小模型纲领中的法诺丛，证明其未归一化凯勒-里奇流的有限时间奇点相关分裂定理、曲率界及切流结构，完善了凯勒-里奇流的奇点理论。
We study the behaviour of the Ricci Yang-Mills flow for U(1) bundles on surfaces. We show that existence for the flow reduces to a bound on the isoperimetric constant. In the presence of such a bound, we show that on $S^2$, if the bundle is nontrivial, the flow exists for all time. For higher genus surfaces the flow always exists for all time. The volume normalized flow always exists for all time and converges to a constant scalar curvature metric with the bundle curvature $F$ parallel. Finally, in an appendix we classify all gradient solitons of this flow on surfaces.
Finite-Time Singularities of the Kähler--Ricci Flow on a $\mathbb{C}P^m$-Bundle over a Product of Kähler--Einstein Manifolds
本文研究 Kähler--Ricci 流在 Kähler--Einstein 流形乘积上的 $\mathbb{CP}^m$ 丛上的有限时间奇点，证明所有奇点均为 Type~I 型。
The aim of this paper is to study self-similar solutions to the symplectic cuvature flow on 6-dimensional nilmanifolds. For this purpose, we focus our attention in the family of symplectic Two- and Three-step nilpotent Lie algebras admitting a "minimal compatible metric" and we give a complete classification of these algebras together with their respective metric. Such classification is given by using our generalization of Nikolayevsky's nice basis criterium (arXiv:1301.4949), which will be repeated here in the context of canonical compatible metrics for geometric structures on nilmanifolds, f…
We generalize the ancient solutions of the Ricci flow on certain principal ${\rm SO}(3)$ bundles over compact quaternionic Kähler manifolds constructed by Bakas, Kong, and Ni to certain $RP^3$ fibre bundles over a product of two compact quaternionic Kähler manifolds. The ancient solutions are of Type I, $κ$-noncollapsed, and have positive Ricci curvature. Analogous solutions on even-dimensional non-Kählerian bundle spaces are also constructed.
In this paper we study collapsing sequences M_{i}-> X of Riemannian manifolds with curvature bounded or bounded away from a controlled subset. We introduce a structure over X which in an appropriate sense is dual to the N-structure of Cheeger, Fukaya and Gromov. As opposed to the N-structure, which live over the M_{i} themselves, this structure lives over X and allows for a convenient notion of global convergence as well as the appropriate background structure for doing analysis on X. This structure is new even in the case of uniformly bounded curvature and as an application we give a generali…
For $ρ, v>0$, we say that an $n$-manifold $M$ satisfies local $(ρ,v)$-bound Ricci covering geometry, if Ricci curvature $\text{Ric}_M\ge -(n-1)$, and for all $x\in M$, $\text{vol}(B_ρ(\tilde x))\ge v>0$, where $\tilde x$ is an inverse image of $x$ on the (local) Riemannian universal cover of the $ρ$-ball at $x$. In this paper, we extend the nilpotent fiber bundle theorem of Cheeger-Fukaya-Gromov on a collapsed $n$-manifold $M$ of bounded sectional curvature to $M$ of a local $(ρ,v)$-bound Ricci covering geometry, and $M$ is close to a non-collapsed Riemannian manifold of lower dimension. The n…
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