We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties:
1) They have positive minimal volume entropy.
2) They satisfy a strict version of the Gromov-Hitchin-Thorpe inequality, with a minimal volume entropy term.
3) They nevertheless admit infinitely many distinct smooth structures for which no compatible Einstein metric exists.
We construct series of examples of exotic smooth structures on compact locally symmetric spaces of noncompact type. In particular, we obtain higher rank examples, which do not support Riemannian metric of nonpositive curvature. The examples are obtained by taking the connected sum with an exotic sphere. To detect the change of the smooth structure we use a tangential map from the locally symmetric space its dual compact type twin.
This paper was withdrawn by the author due to an error in the proof of the main result; essentially the parameter R used in the proof may depend on the manifold (M, g), not just on dimension and pinching constant.
In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time $t \in [0, \infty)$ with uniformly bounded sectional curvature. In dimension 4, there exist many compact topological manifolds admitting distinct smooth structures, i.e., exotic smooth structures. Interestingly, in this article, the difference between existence and non-existence of non-singular solutions to …
In a recent article the first three authors proved that in dimension $4m+1$ all homotopy spheres that bound parallelizable manifolds admit Einstein metrics of positive scalar curvature which, in fact, are Sasakian-Einstein. They also conjectured that all such homotopy spheres in dimension $4m-1, m\geq2$ admit Sasakian-Einstein metrics \cite{BGK}, and proved this for the simplest case, namely dimension $7.$ In this paper we describe computer programs that show that this conjecture is also true for 11-spheres and 15-spheres. Moreover, a program is given that determines the partition of the 8610 …
We show that the smooth $4$-manifold $M$ obtained by attaching a $2$-handle to $B^4$ along a certain knot $K\subset \partial B^4$ admits infinitely many absolutely exotic copies $M_n$, $n=0,1,2..$, such that each copy $M_n$ is obtained by attaching $2$-handle to a fixed compact smooth contractible manifold $W$ along the iterates $f^{n}(c)$ of a knot $c\subset \partial W$ by a diffeomorphism $f:\partial W \to \partial W$. This generalizes the example in author's 1991 paper, which corresponds to $n=1$ case.
In this paper, we investigate existence of inequivalent smooth structures on closed smooth non-orientable 4-manifolds building upon results of Akbulut, Cappell-Shaneson, Fintushel-Stern, Gompf, and Stolz. We add to the number of known constructions and provide new examples of exotic manifolds that are obtained as an application of Gluck twists to the standard smooth structure. Inspection of the smooth structure on the oriented 2-covers yields existence results of orientation-reversing exotic free involutions.
We prove that there are infinitely many pairs of homeomorphic non-diffeomorphic smooth 4-manifolds, such that in each pair one manifold admits an Einstein metric and the other does not. We also show that there are closed 4-manifolds with two smooth structures which admit Einstein metrics with opposite signs of the scalar curvature.
In this paper, we are concerned with interactions between isoparametric theory and differential topology. Two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves. Using differential topology, we obtain several results towards the classification problem of isoparametric foliations up to equivalence. In particular, we show that each homotopy $n$-sphere has the "same" isoparametric foliations as the standard sphere $S^n$ has except for $n=4$, reducing the classification problem on homotopy spheres to that on the standard sphere…
We give a short proof of the (known) result that there are no Kaehler structures on exotic tori. This yields a negative solution to a problem posed by Benson and Gordon. W discuss the symplectic version of the problem and analyze results which yield an evidence for the conjecture that there are no symplectic structures on exotic tori.
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.
Model-theoretic aspects of exotic smoothness were studied long ago uncovering unexpected relations to noncommutative spaces and quantum theory. Some of these relations were worked out in detail in later work. An important point in the argumentation was the forcing construction of Cohen but without a direct application to exotic smoothness. In this article we assign the set-theoretic forcing on trees to Casson handles and characterize small exotic smooth $R^4$ from this point of view. Moreover, we show how models in some Grothendieck toposes can help describing such differential structures in d…
We prove that for every natural number k there are simply connected topological four-manifolds which have at leat k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structures become diffeomorphic after connected sum with only one copy of the complex projective plane. We prove that manifolds with these properties cover a large geographical area.
We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie.
We show that if a Fano manifold $M$ is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then $M$ admits a Kähler-Einstein metric. This is a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-Donaldson-Sun, and can be used to obtain new examples of Kähler-Einstein manifolds. We also give analogous results for twisted Kähler-Einstein metrics and Kahler-Ricci solitons.
Smooth metric measure spaces have been studied from the two different perspectives of Bakry-Émery and Chang-Gursky-Yang, both of which are closely related to work of Perelman on the Ricci flow. These perspectives include a generalization of the Ricci curvature and the associated quasi-Einstein metrics, which include Einstein metrics, conformally Einstein metrics, gradient Ricci solitons, and static metrics. In this article, we describe a natural perspective on smooth metric measure spaces from the point of view of conformal geometry and show how it unites these earlier perspectives within a un…
We show that each of the topological 4-manifolds $CP^2#k\bar{CP^2}, for $k = 6, 7$ admits a smooth structure which has an Einstein metric of scalar curvature $s > 0$, a smooth structure which has an Einstein metric with $s < 0$ and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics. We show that there are infinitely many manifolds homeomorphic non-diffeomorphic to $CP^2#5\bar{CP^2}$ which do not admit an Einstein metric. We also exhibit new higher dimensional examples of manifolds carrying Einstein metrics of both positive and negative scalar curvature. The…
Let $(M^n,g)$, $n \ge 4$, be a compact simply-connected Riemannian manifold with nonnegative isotropic curvature. Given $0<l\le L$, we prove that there exists $\eps = \eps (l,L,n)$ satisfying the following: If the scalar curvature $s$ of $g$ satisfies $$ l \le s \le L $$ and the Einstein tensor satisfies $$ | Ric - \frac {s}{n}g | \le \eps$$ then $M$ is diffeomorphic to a symmetric space of compact type. This is a smooth analogue of the result of S. Brendle that a compact Einstein manifold with nonnegative isotropic curvature is isometric to a locally symmetric space.
We study eta-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem for Sasakian geometry to prove the existence of eta-Einstein structures on many different compact manifolds, including exotic spheres. We also relate these results to the existence of Einstein-Weyl structures.
The existence of \emph{weak conical Kähler-Einstein} metrics along smooth hypersurfaces with angle between $0$ and $2π$ is obtained by studying a smooth continuity method and a \emph{local Moser's iteration} technique. In the case of negative and zero Ricci curvature, the $C^0$ estimate is unobstructed; while in the case of positive Ricci curvature, the $C^0$ estimate obstructed by the properness of the \emph{twisted K-Energy}. As soon as the $C^0$ estimate is achieved, the local Moser iteration could improve the \emph{rough bound} on the approximations to a \emph{uniform $C^2$ bound}, thus pr…
We prove the existence of Kahler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties, and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings.
On a smooth metric measure spacetime $(M,g,e^{-f} dvol_g)$, we define a weighted Einstein tensor. It is given in terms of the Bakry-Émery Ricci tensor as a tensor which is symmetric, divergence-free, concomitant of the metric and the density function. We consider the associated vacuum weighted Einstein field equations and show that isotropic solutions have nilpotent Ricci operator. Moreover, the underlying manifold is a Brinkmann wave if it is $2$-step nilpotent and a Kundt spacetime if it is $3$-step nilpotent. More specific results are obtained in dimension $3$, where all isotropic solutions…
In this paper we study fundamental properties of geodesic mappings with respect to the smoothness class of metrics. We show that geodesic mappings preserve the smoothness class of metrics. We study geodesic mappings of Einstein spaces.
In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space $\bar{\mathcal{M}}$ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on $\bar{\mathcal{M}}$ and extends the canonical Weil-Petersson current on the moduli space parametrizing smooth Kahler-Einstein Fano manifolds $\mathcal{M}$. As a consequence, we show that the CM line bundle is nef and big on $\bar{\mathcal{M}}$ and its restriction on $\mathcal{M}$ is ample.
We survey recent results on the existence of Kähler-Einstein metrics on certain smoothable Fano varieties, focusing on the importance of such metrics in the construction of compact algebraic moduli spaces of K-polystable Fano varieties. Moreover, we give some applications and we discuss some natural problems which deserve future investigations.
Smooth Kahler-Einstein metrics have been studied for the past 80 years. More recently, singular Kahler-Einstein metrics have emerged as objects of intrinsic interest, both in differential and algebraic geometry, as well as a powerful tool in better understanding their smooth counterparts. This article is mostly a survey of some of these developments.
In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano Kähler-Einstein manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski open condition and establish the uniqueness for the Gromov-Hausdorff limit for a punctured flat family of Fano Kähler-Einstein manifolds. Based on these, we construct a proper scheme parameterizing the S-equivalent classes of $\QQ$-Gorenstein smoothable, K-semistable Fano varieties, and verify various necessary properties to guarantee that it is a good …
Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces, and they are interesting examples of spherical varieties. We prove that all smooth Fano symmetric varieties with Picard number one admit Kähler-Einstein metrics by using a combinatorial criterion for K-stability of Fano spherical varieties obtained by Delcroix. For this purpose, we present their algebraic moment polytopes and compute the barycenter of each moment polytope with respect to the Duistermaat-Heckman measure.
In this paper, first we consider the existence and non-existence of Einstein metrics on the topological 4-manifolds $3\mathbb{CP}^2 # k \bar{\mathbb{CP}}^2$ (for $k \in {11, 13, 14, 15, 16, 17, 18}$) by using the idea of Răsdeaconu and Şuvaina (2009) and the constructions in Park, Park, and Shin (arXiv:0906.5195v2) and in Park, Park, and Shin (2009). Then, we study the existence or non-existence of non-singular solutions of the normalized Ricci flow on the exotic smooth structures of these topological manifolds by employing the obstruction developed in Ishida (2008).
We characterise $n$th order ODEs for which the space of solutions $M$ is equipped with a particular paraconformal structure in the sense of \cite{BE}, that is a splitting of the tangent bundle as a symmetric tensor product of rank-two vector bundles. This leads to the vanishing of $(n-2)$ quantities constructed from of the ODE. If $n=4$ the paraconformal structure is shown to be equivalent to the exotic ${\cal G}_3$ holonomy of Bryant. If $n=4$, or $n\geq 6$ and $M$ admits a torsion--free connection compatible with the paraconformal structure then the ODE is trivialisable by point or contact t…
