The aim of this thesis is to construct new examples of compact orbifolds $\mathcal{O}^4(Θ)$ which admit a self dual Einstein (SDE) metric of positive scalar curvature $s>0$, with a one-dimensional group of isometries. In particular we want to prove that these examples are different from those described by Boyer, Galicki and Piccinni in \emph{$3-$Sasakian geometry, nilpotents orbits, and exeptional quotients}. We construct explicitly our new examples as quaternion-K$\ddot{\mathrm{a}}$hler reductions of the quaternion K$\ddot{\mathrm{a}}$hler Grassmannian $Gr_4(\mathbb{R}^8)$ by an isometric action of a $3-$torus $T^3_Θ\subset T^4\subset SO(8)$ $\subset Sp(8)$ on the sphere $S^{31}\subset \mathbb{H}^8$, where $Θ$ is an interger $3\times 4$ weight matrix.
We prove that any asymptotically locally Euclidean scalar-flat Kähler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a $k$-dimensional quaternionic vector space by a $(k-1)$-torus. In order to do so, we first prove that any compact anti-self-dual 4-orbifold with positive Euler characteristic whose isometry group contains a 2-torus is conformally equivalent, up to an orbifold covering, to a quaternionic quotient of $k$-dimensional quaternionic projective space by a $(k-1)$-torus.
We proved the convergence of a sequence of 2 dimensional comapct Kahler-Einstein orbifolds with rational quotient singularities and with some uniform bounds on the volumes and on the Euler characteristics of our orbifods to a Kahler-Einstein 2-dimensional orbifold. Our limit orbifold can have worse singularities than the orbifolds in our sequence. We will also derive some estimates on the norms of the sections of plurianticanonical bundles of our orbifolds in the sequence that we are considering and our limit orbifold.
We construct a new family of compact orbifolds with a positive self dual Einstein metric and a one-dimensional group of isometries. Together with another known family, these examples classify all 4-dimensional orbifolds that are quaternion Kaehler quotients by a torus of real Grassmannians.
We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal structure. A compact anti-self-dual orbifold can also be constructed by adding a point at infinity to an asymptotically locally Euclidean (ALE) scalar-flat Kähler orbifold. We use this observation to classify ALE scalar-flat Kähler 4-orbifolds whose isometry group contain a 2-torus.
Consider an Einstein orbifold $(M_0,g_0)$ of real dimension $2n$ having a singularity with orbifold group the cyclic group of order $n$ in ${\rm{SU}}(n)$ which is generated by an $n$th root of unity times the identity. Existence of a Ricci-flat Kähler ALE metric with this group at infinity was shown by Calabi. There is a natural "approximate" Einstein metric on the desingularization of $M_0$ obtained by replacing a small neighborhood of the singular point of the orbifold with a scaled and truncated Calabi metric. In this paper, we identify the first obstruction to perturbing this approximate E…
An orbifold version of the Hitchin-Thorpe inequality is used to prove that certain weighted projective spaces do not admit orbifold Einstein metrics. Also, several estimates for the orbifold Yamabe invariants of weighted projective spaces are proved.
A theorem of Anderson and Bando-Kasue-Nakajima from 1989 states that to compactify the set of normalized Einstein metrics with a lower bound on the volume and an upper bound on the diameter in the Gromov-Hausdorff sense, one has to add singular spaces called Einstein orbifolds, and the singularities form as blow-downs of Ricci-flat ALE spaces. This raises some natural issues, in particular: can all Einstein orbifolds be Gromov-Hausdorff limits of smooth Einstein manifolds? Can we describe more precisely the smooth Einstein metrics close to a given singular one? In this first paper, we prove th…
A construction of Kaehler-Einstein metrics using Galois coverings, studied by Arezzo-Ghigi-Pirola, is generalized to orbifolds. By applying it to certain orbifold covers of P^n which are trivial set theoretically, one obtains new Einstein metrics on odd-dimensional spheres. The method also gives Kaehler-Einstein metrics on degree 2 Del Pezzo surfaces with A_1 or A_2-singularities.
We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4-dimensional ALF hyperkähler metrics. We apply our construction to asymptotically conical G2 metrics arising from self-dual Einstein 4-orbifolds with positive scalar curvature. As …
We prove that the identity component of the holomorphic isometry group of a Sasaki-Einstein metric is the identity component of a maximal compact subgroup of its automorphism group.
We give an explicit local classification of conformally equivalent but oppositely oriented Kaehler metrics on a 4-manifold which are toric with respect to a common 2-torus action. In the generic case, these structures have an intriguing local geometry depending on a quadratic polynomial and two arbitrary functions of one variable, these two functions being explicit degree 4 polynomials when the Kaehler metrics are extremal (in the sense of Calabi). One motivation for and application of this result is an explicit local description of Einstein 4-manifolds which are hermitian with respect to eith…
In this paper, we construct families of nonisometric hyperbolic orbifolds that contain the same isometry classes of nonflat totally geodesic subspaces. The main tool is a variant of the well-known Sunada method for constructing length-isospectral Riemannian manifolds that handles totally geodesic submanifolds of multiple codimensions simultaneously.
In this note, we prove that any non-collapsing and compact Gromov-Hausdorff limit of Kahler-Einstein manifolds is either smooth or is orbifold outside a subvariety of complex codimension at least 3.
We prove the following result: Let $(X,g_0)$ be a complete, connected 4-manifold with uniformly positive isotropic curvature and with bounded geometry. Then there is a finite collection $\mathcal{F}$ of manifolds of the form $\mathbb{S}^3 \times \mathbb{R} /G$, where $G$ is a discrete subgroup of the isometry group of the round cylinder $\mathbb{S}^3\times \mathbb{R}$ on which $G$ acts freely, such that $X$ is diffeomorphic to a possibly infinite connected sum of $\mathbb{S}^4,\mathbb{RP}^4$ and members of $\mathcal{F}$. This extends recent work of Chen-Tang-Zhu and Huang. We also extend the a…
For a holomorphic vector bundle over a compact Kähler orbifold, the slope stability of the bundle is shown to be equivalent to the existence of a Hermitian-Einstein metric or to the properness of a certain functional introduced by Donaldson.
We study the size of the isometry group Isom(M, g) of Riemannian manifolds (M, g) as g varies. For M not admitting a circle action, we show that the order of Isom(M, g) can be universally bounded in terms of the bounds on Ricci curvature, diameter, and injectivity radius of M. This generalizes results known for negative Ricci curvature to all manifolds. More generally we establish a similar universal bound on the index of the deck group pi_1(M) in the isometry group Isom(X, g) of the universal cover X in the absence of suitable actions by connected groups. We apply this to characterize locally…
In this short note, we investigate the existence of orbifold Kähler-Einstein metrics on toric varieties. In particular, we show that every $\mathbb{Q}$-factorial normal projective toric variety allows an orbifold Kähler-Einstein metric. Moreover, we characterize the K-stability of $\mathbb{Q}$-factorial toric pairs of Picard number one in terms of the log Cox ring and the universal orbifold cover.
We consider 3+1 rotationally symmetric Lorentzian Einstein spacetime manifolds with $Λ>0$ and reduce the equations to 2+1 Einstein equations coupled to `shifted' wave maps. Subsequently, we prove various (explicit) positive mass-energy theorems. No smallness is assumed.
We establish connections between contact isometry groups of certain contact manifolds and compactly supported symplectomorphism groups of their symplectizations. We apply these results to investigate the space of symplectic embeddings of balls with a single conical singularity at the origin. Using similar ideas, we also prove the longstanding expected result that the space of Lagrangian $\RR P^2$ in $T^*\RR P^2$ is weakly contractible.
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 prove that any compact selfdual Einstein 4-orbifold of positive scalar curvature whose isometry group contains a 2-torus is, up to an orbifold covering, a quaternion Kaehler quotient of (k-1)-dimensional quaternionic projective space by a (k-2)-torus for some $k\geq 2$. We also obtain a topological classification in terms of the intersection form of the 4-orbifold.
Sixty years ago, S. B. Myers and N. E. Steenrod ({\it Ann. of Math.} {\bf 40} (1939), 400-416) showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. Recently A. V. Bagaev and N. I. Zhukova ({\it Siberian Math. J.} {\bf 48} (2007), 579-592) proved the same result for a Riemannian orbifold. In this paper, we firstly show that the isometry group of a Riemannian manifold $M$ with boundary has dimension at most ${1/2} \dim M (\dim M-1)$. Then we completely classify such Riemannian manifolds with boundary that their isometry groups attain the precedin…
We find a new obstruction for a real Einstein 4-orbifold with an A1-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction is a curvature condition at the singular point. For asymptotically hyperbolic metrics, with boundary at infinity a conformal metric, we prove that if the obstruction vanishes, one can desingularize Einstein orbifolds with such singularities. The Dirichlet problem consists in finding Einstein metrics with given conformal infinity on the boundary: we prove that our obstruction defines a wall in the space of conformal metrics on the boundary, and that all t…
We show that $\scriptstyle{#9(S^2\times S^3)}$ admits an 8-dimensional complex family of inequivalent non-regular Sasakian-Einstein structures. These are the first known Einstein metrics on this 5-manifold. In particular, the bound $\scriptstyle{b_2(M)\leq8}$ which holds for any regular Sasakian-Einstein $\scriptstyle{M}$ does not apply to the non-regular case. We also discuss the failure of the Hitchin-Thorpe inequality in the case of 4-orbifolds and describe the orbifold version.
We find new obstructions to the desingularization of compact Einstein orbifolds by smooth Einstein metrics. These new obstructions, specific to the compact situation, raise the question of whether a compact Einstein $4$-orbifold which is limit of Einstein metrics bubbling out Eguchi-Hanson metrics has to be Kähler. We then test these obstructions to discuss if it is possible to produce a Ricci-flat but not Kähler metric by the most promising desingularization configuration proposed by Page in 1981. We identify $84$ obstructions which, once compared to the $57$ degrees of freedom, indicate that…
The requirement that a (non-Einstein) Kähler metric in any given complex dimension $m>2$ be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local biholomorphic-isometry types of such metrics depend, for each $m>2$, on three real parameters along with an arbitrary Kähler-Einstein metric $h$ in complex dimension $m-1$. We provide an explicit description of all these local-isometry types, for any given $h$. That result is derived from a more general local classification theorem for metrics admitting functions we…
In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold is the result of a gluing-perturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted Hölder spaces, in which the analysis of the present paper takes place. The description of Einstein metrics as the result of a gluing-perturbation procedure also sheds light on the local structure of the moduli space of Einstein metrics near its boundary. More importantly here, we…
The purpose of this article is to study the existence and uniqueness of quasi-Einstein structures on $3$-dimensional homogeneous Riemannian manifolds. To this end, we use the eight model geometries for 3-dimensional manifolds identified by Thurston. First, we present here a complete description of quasi-Einstein metrics on $3$-dimensional homogeneous manifolds with isometry group of dimension $4.$ In addition, we shall show the absence of such gradient structure on $Sol^3,$ which has $3$-dimensional isometry group. Moreover, we prove that Berger's spheres carry a non-trivial quasi-Einstein str…
We show that any compact convex simple lattice polytope is the moment polytope of a Kähler-Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang-Zhu Theorem \cite{WZ} giving the existence of a Kähler-Ricci soliton on any toric monotone manifold on any compact convex simple labelled polytope satisfying the combinatoric condition corresponding to monotonicity. We obtain that any compact convex simple polytope $P\subset \bR^n$ admits a set of inward normals, unique up to dilatation, such that there exists a symplectic potential satisfying the Guillemin boundary condi…
Let (M,g) be a compact oriented Einstein 4-manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M,g) is CP2, equipped with its standard Fubini-Study metric.
Seiberg-Witten theory is used to obtain new obstructions to the existence of Einstein metrics on 4-manifolds with conical singularities along an embedded surface. In the present article, the cone angle is required to be of the form 2(pi)/p, where p is a positive integer, but we conjecture that similar results will also hold in greater generality.
We provide an explicit resolution of the existence problem for extremal Kaehler metrics on toric 4-orbifolds M with second Betti number b2(M)=2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi et al.). Furthermore, in this case, the extremal Kaehler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kaehler metric, which turns out to be extremal as well. These results…
In this paper, we study the behavior of Ricci flows on compact orbifolds with finite singularities. We show that Perelman's pseudolocality theorem also holds on orbifold Ricci flow. Using this property, we obtain a weak compactness theorem of Ricci flows on orbifolds under some natural technical conditions. This generalizes the corresponding theorem on manifolds. As an application, we can use Kähler Ricci flow to find new Kähler Einstein metrics on some orbifold Fano surfaces. For example, if $Y$ is a cubic surface with only one ordinary double point or $Y$ is an orbifold Fano surface with deg…
We calculate a wall crossing formula for 4-dimensional Poincare-Einstein metrics, through a wall made of orbifold Poincare-Einstein metrics with A1 singularities. This is based on a formalism which enables to deal with higher order terms of the Einstein equation in this setting. Some other consequences are deduced.
Let (M,h) be a compact 4-dimensional Einstein manifold, and suppose that h is Hermitian with respect to some complex structure J on M. Then either (M,J,h) is Kaehler-Einstein, or else, up to rescaling and isometry, it is one of the following two exceptions: the Page metric on CP2 # (-CP2), or the Einstein metric on CP2 # 2 (-CP2) constructed in Chen-LeBrun-Weber.
The aim of this paper is to construct infinitely many families of Einstein metrics on the connected sums of arbitrary number of copies of $S^2\times S^3$. We realize these 5-manifolds as total spaces of Seifert bundles over Del Pezzo orbifolds. A Kähler--Einstein metric on the Del Pezzo orbifold is then lifted to an Einstein metric using the Kobayashi--Boyer--Galicki method.
Cartan-Hadamard manifold is a simply connected Riemannian manifold with non-positive sectional curvature. In this article, we have proved that a Cartan-Hadamard manifold satisfying steady gradient Ricci soliton with the integral condition of potential function is isometric to the Euclidean space. Next we have proved a compactness theorem for gradient shrinking Ricci soliton satisfying some scalar curvature condition. Finally, we have showed that a gradient expanding Ricci soliton with linear volume growth and positive potential function is an Einstein manifold.
In this paper we prove that generic small partial smoothings of Kahler-Einstein (KE) Del Pezzo orbifolds with only nodal singularities, and with no non-zero holomorphic vector fields, admit orbifold KE metrics which are close in the Gromov-Hausdorff sense to the original KE metric.
Negative Sasakian manifolds, where the first Chern class of the contact subbundle is a torsion class, can be viewed as Seifert-$S^1$ bundles where the base orbifold has an ample orbifold canonical class. We use this framework to settle completely an open problem formulated by C.Boyer and K.Galicki which asks whether or not arbitrary connected sums of $S^2\times S^3$ are negative Sasakian manifolds. As a consequence of the affirmative answer to this problem, there exists so-called Sasaki eta-Einstein and Lorentzian Sasaki-Einstein metrics on all of these five-manifolds and moreover all of these…
This work addresses the questions: (i) Among all left-invariant Riemannian metrics on a given Lie group, is there any whose isometry group or isometry algebra contain that of all others? (ii) Do expanding left-invariant Ricci solitons exhibit such maximal symmetry? Question (i) is addressed both for semisimple and for solvable Lie groups. Building on previous work of the authors on Einstein metrics, a complete answer is given to (ii): expanding homogeneous Ricci solitons have maximal isometry algebras although not always maximal isometry groups. As a consequence of the tools developed to addre…
Recently, it was shown that Einstein solvmanifolds have maximal symmetry in the sense that their isometry groups contain the isometry groups of any other left-invariant metric on the given Lie group. Such a solvable Lie group is necessarily non-unimodular. In this work we consider unimodular solvable Lie groups and prove that there is always some metric with maximal symmetry. Further, if the group at hand admits a Ricci soliton, then it is the isometry group of the Ricci soliton which is maximal.
Recently, Atiyah and LeBrun proved versions of the Gauss-Bonnet and Hirzebruch signature Theorems for metrics with edge-cone singularities in dimension four, which they applied to obtain an inequality of Hitchin-Thorpe type for Einstein edge-cone metrics. Interestingly, many natural examples of edge-cone metrics in dimension four are anti-self-dual (or self-dual depending upon choice of orientation). On such a space there is an important elliptic complex called the anti-self-dual deformation complex, whose index gives crucial information about the local structure of the moduli space of anti-se…
We prove the following result: Let $(X,g_0)$ be a complete, connected 4-manifold with uniformly positive isotropic curvature and with bounded geometry. Then there is a finite collection $\mathcal{F}$ of manifolds of the form $\mathbb{S}^3 \times \mathbb{R} /G$, where $G$ is a fixed point free discrete subgroup of the isometry group of the standard metric on $\mathbb{S}^3\times \mathbb{R}$, such that $X$ is diffeomorphic to a (possibly infinite) connected sum of copies of $\mathbb{S}^4,\mathbb{RP}^4$ and/or members of $\mathcal{F}$. This extends recent work of Chen-Tang-Zhu and Huang. We also e…
We investigate compact Kahler manifolds, which are acted on by a semisimple compact Lie group G of isometries with one hypersurface orbit. In case of ordinary action and projectable complex structure, we set up a one to one correspondence between such manifolds and abstract models. The Ricci tensor is then computed and we fully characterize the Kahler-Einstein manifolds with an ordinary cohomogeneity one action and projectable complex structure.
In this paper, we show that the uniform L^{4}-bound of the transverse Ricci curvature along the Sasaki-Ricci flow on a compact quasi-regular Sasakian (2n+1)-manifold M of general type. As an application, any solution of the normalized Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular Sasaki η-Einstein metric on the transverse canonical model M_{can} of M if n is less than or equal to 3. In particular for n equal to 2, M_{can} is a S^{1}-orbibundle over the unique Keahler-Einstein orbifold surface (Z_{can},ω_{KE}) with finite point orbifold singularities. The floati…
We prove the existence of an abundance of new Einstein metrics on odd dimensional spheres including exotic spheres, many of them depending on continuous parameters. The number of families as well as the number of parameter grows double exponentially with the dimension. Our method of proof uses Brieskorn-Pham singularities to realize spheres (and exotic spheres) as circle orbi-bundles over complex algebraic orbifolds, and lift a Kaehler-Einstein metric from the orbifold to a Sasakian-Einstein metric on the sphere.
The analysis of holomorphic sections of high powers $L^N$ of holomorphic ample line bundles $L\to M$ over compact Kähler manifolds has been widely applied in complex geometry and mathematical physics. The Tian-Yau-Zelditch's asymptotic expansion of the Szegö kernel of a circle bundle plays an important role in Kähler-Einstein geometry. We generalize the expansion for compact Kähler orbifolds with only finite isolated singularities and analyze the behavior of the expansion near the singularities.
On a given compact complex manifold or orbifold $(M,J)$, we study the existence of Hermitian metrics $\tilde g$ in the conformal classes of Kähler metrics on $(M,J)$, such that the Ricci tensor of $\tilde g$ is of type $(1,1)$ with respect to the complex structure, and the scalar curvature of $\tilde g$ is constant. In real dimension $4$, such Hermitian metrics provide a Riemannian counter-part of the Einstein--Maxwell (EM) equations in general relativity, and have been recently studied in \cite{ambitoric1, LeB0, LeB, KTF}. We show how the existence problem of such Hermitian metrics (which we …
Let $X$ be a Hadamard manifold, and $Γ$ a non-elementary discrete group of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/Γ$ to the behavior of the Poincar{é} series of $Γ$. Precisely, the aim of this paper is to extend the so-called theorem of Hopf-Tsuji-Sullivan -- well-known for manifolds of pinched negative curvature -- to the framework of rank one orbifolds. Moreover, we derive some important properties for $Γ$-invariant conformal densities supported on the geometric limit set of $Γ$.
We embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of Kodaira embedding. Dividing by the (non-reductive) automorphisms of weighted projective space then formally gives a moduli space of orbifolds. We show how to express this as a reductive quotient and so a GIT problem, thus defining a notion of stability for orbifolds. We then prove an orbifold version of Donaldson's theorem: the existence of an orbifold Kahler metric of constant scalar curvature implies K-semistability. By extending the notion of slope stability to orbifolds we there…
We show that on smooth minimal surfaces of general type, the Kähler-Ricci flow starting at any initial Kähler metric converges in the Gromov-Hausdorff sense to a Kähler-Einstein orbifold surface. In particular, the diameter of the evolving metrics is uniformly bounded for all time and the Kähler-Ricci flow contracts all the holomorphic spheres with $(-2)$ self-intersection number to isolated orbifold points. Our estimates do not require a priori the existence of an orbifold Kähler-Einstein metric on the canonical model.
We apply the Berglund-Hübsch transpose rule from BHK mirror symmetry to show that to an $n-1$-dimensional Calabi-Yau orbifold in weighted projective space defined by an invertible polynomial, we can associate four (possibly) distinct Sasaki manifolds of dimension $2n+1$ which are $n-1$-connected and admit a metric of positive Ricci curvature. We apply this theorem to show that for a given K3 orbifold, there exists four seven dimensional Sasakian manifolds of positive Ricci curvature, two of which are actually Sasaki-Einstein.
