Diffeomorphism Types Research Articles

On the intersection form of fillings

AbstractWe prove, by an ad hoc method, that exact fillings with vanishing rational first Chern class of flexibly fillable contact manifolds have unique integral intersection forms. We appeal to the special Reeb dynamics (stronger than ADC in [Lazarev, Geom. Funct. Anal. 30 (2020), no. 1, 188–254]) on the contact boundary, while a more systematic approach working for general ADC manifolds is developed independently by Eliashberg, Ganatra and Lazarev. We also discuss cases where the vanishing rational first Chern class assumption can be removed. We derive the uniqueness of diffeomorphism types of exact fillings of certain flexibly fillable contact manifolds and obstructions to contact embeddings, which are not necessarily exact.

On fillings of contact links of quotient singularities

AbstractWe study several aspects of fillings for links of general isolated quotient singularities using Floer theory, including co‐fillings, Weinstein fillings, strong fillings, exact fillings and exact orbifold fillings, focusing on the non‐existence of exact fillings of contact links of isolated terminal quotient singularities. We provide an extensive list of isolated terminal quotient singularities whose contact links are not exactly fillable, including for , which settles a conjecture of Eliashberg, quotient singularities from general cyclic group actions and finite subgroups of , and all terminal quotient singularities in complex dimension 3. We also obtain uniqueness of the orbifold diffeomorphism type of exact orbifold fillings of contact links of some isolated terminal quotient singularities.

The fixed point set of the inverse involution on a Lie group

The inverse involution on a Lie group $G$ is the periodic $2$ transformation $\gamma $ that sends each element $g\in G$ to its inverse $g^{-1}$. The variety of the fixed point set $\Fix(\gamma )$ is of importance for the relevances with Morse function on the Lie group $G$, and the $G$-representations of the cyclic group $\mathbb{Z}_{2}$. In this paper we develop an approach to calculate the diffeomorphism types of the fixed point sets $\Fix(\gamma)$ for the simple Lie groups.

ON THE MILNOR FIBRATION OF CERTAIN NEWTON DEGENERATE FUNCTIONS

AbstractIt is well known that the diffeomorphism type of the Milnor fibration of a (Newton) nondegenerate polynomial function f is uniquely determined by the Newton boundary of f. In the present paper, we generalize this result to certain degenerate functions, namely we show that the diffeomorphism type of the Milnor fibration of a (possibly degenerate) polynomial function of the form $f=f^1\cdots f^{k_0}$ is uniquely determined by the Newton boundaries of $f^1,\ldots , f^{k_0}$ if $\{f^{k_1}=\cdots =f^{k_m}=0\}$ is a nondegenerate complete intersection variety for any $k_1,\ldots ,k_m\in \{1,\ldots , k_0\}$ .

On the topology of real Lagrangians in toric symplectic manifolds

We explore the topology of real Lagrangian submanifolds in a toric symplectic manifold which come from involutive symmetries on its moment polytope. We establish a real analog of the Delzant construction for those real Lagrangians, which says that their diffeomorphism type is determined by combinatorial data. As an application, we realize all possible diffeomorphism types of connected real Lagrangians in toric symplectic del Pezzo surfaces.

Lorentzian Gromov–Hausdorff theory and finiteness results

Cheeger–Gromov finiteness results, asserting that there are only finitely many diffeomorphism types of manifolds satisfying certain geometric bounds, feature among the most prominent results in Riemannian geometry. To transplant those into Lorentzian geometry, one could use a functor between a Lorentzian and a Riemannian category, which, however, can be shown not to exist if the former contains Minkowski space and its isometries. Here, we construct a functor from a restricted category of Lorentzian manifolds-with-boundary (regions between two Cauchy surfaces) to a category of Riemannian manifolds-with-boundary that preserves geometric bounds and obtain, as a corollary, the first known Lorentzian Cheeger–Gromov type finiteness result.

Topology and Curvature of Isoparametric Families in Spheres

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik–Schnirelmann category. This part extends substantially the results of Wang (J Differ Geom 27:55–66, 1988). The second part is concerned with their curvatures; more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

On the symplectic fillings of standard real projective spaces

We prove, in a geometric way, that the standard contact structure on \({{\mathbb {R}}}{{\mathbb {P}}}^{2n-1}\) is not Liouville fillable for \(n \ge 3\) and odd. We also prove for all n that semipositive fillings of such contact structures are always simply connected. Finally, we give yet another proof of the Eliashberg–Floer–McDuff theorem on the diffeomorphism type of the symplectically aspherical fillings of the standard contact structure on \(S^{2n-1}\).

Diffeomorphism type via aperiodicity in Reeb dynamics

We characterise boundary shaped disc like neighbourhoods of certain isotropic submanifolds in terms of aperiodicity of Reeb flows. We prove uniqueness of homotopy and diffeomorphism type of such contact manifolds assuming non-existence of short periodic Reeb orbits.

Small curvature concentration and Ricci flow smoothing

We show that a complete Ricci flow of bounded curvature which begins from a manifold with a Ricci lower bound, local entropy bound, and small local scale-invariant integral curvature control will have global point-wise curvature control at positive times which only depends on the initial almost Euclidean structure. As applications, we use the Ricci flows to study the diffeomorphism type of manifolds and the regularity of Gromov-Hausdorff limit of manifolds with small curvature concentration.

Umbilical submanifolds of [formula omitted

In this article we complete the classification of the umbilical submanifolds of a Riemannian product of two space forms, addressing the case of a conformally flat product Hk×Sn−k+1, which has not been covered in previous works on the subject. We show that there exists precisely a p-parameter family of congruence classes of umbilical submanifolds of Hk×Sn−k+1 with substantial codimension p, which we prove to be at most min{k+1,n−k+2}. We study more carefully the cases of codimensions one and two and exhibit, respectively, a one-parameter family and a two-parameter family (together with three extra one-parameter families) that contain precisely one representative of each congruence class of such submanifolds. In particular, this yields another proof of the classification of all (congruence classes of) umbilical submanifolds of Sn×R, and provides a similar classification for the case of Hn×R. We determine all possible topological types, actually, diffeomorphism types, of a complete umbilical submanifold of Hk×Sn−k+1. We also show that umbilical submanifolds of the product model of Hk×Sn−k+1 can be regarded as rotational submanifolds in a suitable sense, and explicitly describe their profile curves when k=n. As a consequence of our investigations, we prove that every conformal diffeomorphism of Hk×Sn−k+1 onto itself is an isometry.

The Torelli theorem for gravitational instantons

Abstract We give a short proof of the Torelli theorem for $ALH^*$ gravitational instantons using the authors’ previous construction of mirror special Lagrangian fibrations in del Pezzo surfaces and rational elliptic surfaces together with recent work of Sun-Zhang. In particular, this includes an identification of 10 diffeomorphism types of $ALH^*_b$ gravitational instantons.

Finite Diffeomorphism Types of Four Dimensional Ricci Flow with Bounded Scalar Curvature

Finite Diffeomorphism Types of Four Dimensional Ricci Flow with Bounded Scalar Curvature

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.

Homological stability for moduli spaces of disconnected submanifolds, I

A well-known property of unordered configuration spaces of points (in an open, connected manifold) is that their homology stabilises as the number of points increases. We generalise this result to moduli spaces of submanifolds of higher dimension, where stability is with respect to the number of components having a fixed diffeomorphism type and isotopy class. As well as for unparametrised submanifolds, we prove this also for partially-parametrised submanifolds -- where a partial parametrisation may be thought of as a superposition of parametrisations related by a fixed subgroup of the mapping class group. In a companion paper (arXiv:1807.07558), this is further generalised to submanifolds equipped with labels in a bundle over the embedding space, from which we deduce corollaries for the stability of diffeomorphism groups of manifolds with respect to parametrised connected sum and addition of singularities.

Contraction centers in families of hyperkähler manifolds

We study the exceptional loci of birational (bimeromorphic) contractions of a hyperk\"ahler manifold $M$. Such a contraction locus is the union of all minimal rational curves in a collection of cohomology classes which are orthogonal to a wall of the K\"ahler cone. Homology classes which can possibly be orthogonal to a wall of the K\"ahler cone of some deformation of $M$ are called MBM classes. We prove that all MBM classes of type (1,1) can be represented by rational curves, called MBM curves. All MBM curves can be contracted on an appropriate birational model of $M$, unless $b_2(M) \leq 5$. When $b_2(M)>5$, this property can be used as an alternative definition of an MBM class and an MBM curve. Using the results of Bakker and Lehn, we prove that the diffeomorphism type of a contraction locus remains stable under all deformations for which these classes remains of type (1,1), unless the contracted variety has $b_2\leq 4$. Moreover, these diffeomorphisms preserve the MBM curves, and induce biholomorphic maps on the contraction fibers, if they are normal.

Families of monotone Lagrangians in Brieskorn\u2013Pham hypersurfaces

We present techniques, inspired by monodromy considerations, for constructing compact monotone Lagrangians in certain affine hypersurfaces, chiefly of Brieskorn–Pham type. We focus on dimensions 2 and 3, though the constructions generalise to higher ones. The techniques give significant latitude in controlling the homology class, Maslov class and monotonicity constant of the Lagrangian, and a range of possible diffeomorphism types; they are also explicit enough to be amenable to calculations of pseudo-holomorphic curve invariants. Applications include infinite families of monotone Lagrangian S^1 times Sigma _g in {mathbb {C}}^3, distinguished by soft invariants for any genus g ge 2; and, for fixed soft invariants, a range of infinite families of Lagrangians in Brieskorn–Pham hypersurfaces. These are generally distinct up to Hamiltonian isotopy. In specific cases, we also set up well-defined counts of Maslov zero holomorphic annuli, which distinguish the Lagrangians up to compactly supported symplectomorphisms. Inter alia, these give families of exact monotone Lagrangian tori which are related neither by geometric mutation nor by compactly supported symplectomorphisms.

Infinitely many new families of complete cohomogeneity one G$_2$-manifolds: G$_2$ analogues of the Taub–NUT and Eguchi–Hanson spaces

We construct infinitely many new 1-parameter families of simply connected complete non-compact G _2 -manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter values: at one special value we obtain a unique member of each family with asymptotically conical (AC) geometry; on approach to the other special parameter value the family of metrics collapses to an AC Calabi–Yau 3-fold. Our infinitely many new diffeomorphism types of AC G _2 -manifolds are particularly noteworthy: previously the three examples constructed by Bryant and Salamon in 1989 furnished the only known simply connected AC G _2 -manifolds. We also construct a closely related conically singular G _2 -holonomy space: away from a single isolated conical singularity, where the geometry becomes asymptotic to the G _2 -cone over the standard nearly Kähler structure on the product of a pair of 3-spheres, the metric is smooth and it has ALC geometry at infinity. We argue that this conically singular ALC G2-space is the natural G _2 analogue of the Taub–NUT metric in 4-dimensional hyperKähler geometry and that our new AC G _2 -metrics are all analogues of the Eguchi–Hanson metric, the simplest ALE hyperKähler manifold. Like the Taub–NUT and Eguchi–Hanson metrics, all our examples are cohomogeneity one, i.e. they admit an isometric Lie group action whose generic orbit has codimension one.

Symplectic fillings of asymptotically dynamically convex manifolds I

We consider exact fillings with vanishing first Chern class of asymptotically dynamically convex (ADC) manifolds. We construct two structure maps on the positive symplectic cohomology and prove that they are independent of the filling for ADC manifolds. The invariance of the structure maps implies that the vanishing of symplectic cohomology and the existence of symplectic dilations are properties independent of the filling for ADC manifolds. Using them, various topological applications on symplectic fillings are obtained, including the uniqueness of diffeomorphism types of fillings for many contact manifolds. We use the structure maps to define the first symplectic obstructions to Weinstein fillability. In particular, we show that for all dimension $4k+3, k\ge 1$, there exist infinitely many contact manifolds that are exactly fillable, almost Weinstein fillable but not Weinstein fillable. The invariance of the structure maps generalizes to strong fillings with vanishing first Chern class. We show that any strong filling with vanishing first Chern class of a class of manifolds, including $(S^{2n-1},\xi_{std}), \partial(T^*L \times \mathbb{C}^n)$ with $L$ simply connected, must be exact and have unique diffeomorphism type. As an application of the proof, we show that the existence of symplectic dilation implies uniruledness. In particular any affine exotic $\mathbb{C}^n$ with non-negative log Kodaira dimension is a symplectic exotic $\mathbb{C}^{n}$.

Moduli space of metrics of nonnegative sectional or positive Ricci curvature on homotopy real projective spaces

We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy R P 5 \mathbb {R} P^5 has infinitely many path components. We also show that in each dimension 4 k + 1 4k+1 there are at least 2 2 k 2^{2k} homotopy R P 4 k + 1 \mathbb {R} P^{4k+1} ’s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions 4 k + 3 ≥ 7 4k+3\geq 7 .