Cheeger Gromov Research Articles

Bounds on Cheeger–Gromov invariants and simplicial complexity of triangulated manifolds

Abstract We show the existence of linear bounds on Wall 𝜌-invariants of PL manifolds, employing a new combinatorial concept of 𝐺-colored polyhedra. As an application, we show how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with 𝑉 simplices and the fundamental group of Z n \mathbb{Z}_{n} grows in 𝑉. Furthermore, we count the number of homotopy lens spaces with bounded geometry in 𝑉. Similarly, we give new linear bounds on Cheeger–Gromov 𝜌-invariants of PL manifolds endowed with a faithful representation also. A key idea is to construct a cobordism with a linear complexity whose boundary is π 1 \pi_{1} -injectively embedded, using relative hyperbolization. As an application, we study the complexity theory of high-dimensional lens spaces. Lastly, we show the density of 𝜌-invariants over manifolds homotopy equivalent to a given manifold for certain fundamental groups. This implies that the structure set is not finitely generated.

Bach flow of simply connected nilmanifolds

Abstract The Bach flow is a fourth-order geometric flow defined on four-manifolds. For a compact manifold, it is the negative gradient flow for the L 2-norm of the Weyl curvature. In this paper, we study the Bach flow on four-dimensional simply connected nilmanifolds whose Lie algebra is indecomposable. We show that the Bach flow beginning at an arbitrary left invariant metric exists for all positive times and after rescaling converges in the pointed Cheeger–Gromov sense to an expanding Bach soliton which is non-gradient. Combining our results with previous results of Helliwell gives a complete description of the Bach flow on simply connected nilmanifolds.

On the pluriclosed flow on Oeljeklaus–Toma manifolds

AbstractWe investigate the pluriclosed flow on Oeljeklaus–Toma manifolds. We parameterize left-invariant pluriclosed metrics on Oeljeklaus–Toma manifolds, and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution $\omega _t$ which once normalized collapses to a torus in the Gromov–Hausdorff sense. Moreover, the lift of $\tfrac {1}{1+t}\omega _t$ to the universal covering of the manifold converges in the Cheeger–Gromov sense to $(\mathbb H^s\times \mathbb C^s, \tilde {\omega }_{\infty })$ , where $\tilde {\omega }_{\infty }$ is an algebraic soliton.

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.

Enhanced Bounds for rho-invariants for both general and spherical 3-manifolds

We establish enhanced bounds on Cheeger–Gromov [Formula: see text]-invariants for general 3-manifolds and yet stronger bounds for special classes of 3-manifold. As key ingredients, we construct chain null-homotopies whose complexity is linearly bounded by its boundary. This result can be regarded as an algebraic topological analogue of Gromov’s conjecture for quantitative topology. The author hopes for applications to various fields including the smooth knot concordance group, quantitative topology and complexity theory.

Boju–Funar type theorems via m-Bakry–Émery and m-modified Ricci curvatures

We establish some Boju–Funar type compactness criteria for complete Riemannian manifolds via [Formula: see text]-Bakry–Émery and [Formula: see text]-modified Ricci curvatures assuming that [Formula: see text]-Bakry–Émery and [Formula: see text]-modified Ricci curvatures tend slowly to zero as the distance from a fixed point goes to infinity. Our results are generalizations of Cheeger–Gromov–Taylor type compactness criteria for complete Riemannian manifolds via [Formula: see text]-Bakry–Émery and [Formula: see text]-modified Ricci curvatures established by Y. Soylu and the author.

The Nirenberg problem of prescribed Gauss curvature on $S^2$

We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on S^{2} conformal to the round metric. A key tool is to employ the smooth Cheeger–Gromov compactness theorem to obtain general and essentially sharp a priori estimates for Gauss curvatures K contained in naturally defined stable regions. We prove that in such stable regions, the map u \to K_{g} , g = e^{2u}g_{+1} is a proper Fredholm map with well-defined degree on each component. This leads to a number of new existence and non-existence results. We also present a new proof and generalization of the Moser theorem on Gauss curvatures of even conformal metrics on S^{2} . In contrast to previous work, the work here does not use any of the Sobolev-type inequalities of Trudinger–Moser–Aubin–Onofri.

Comparison theorems on Riemannian–Finsler manifolds with curvature quadratic decay and their applications

We establish some comparison theorems on Finsler manifolds with curvature quadratic decay. As their applications, we obtain some optimal Cheeger–Gromov–Taylor type compact theorems, volume growth and Mckean type estimate for the first Dirichlet eigenvalue for such manifolds. Although we present the results for Finsler manifolds, they are all new results for Riemannian manifolds.

Positive Hermitian curvature flow on complex 2-step nilpotent Lie groups

We study the positive Hermitian curvature flow of left-invariant metrics on complex 2-step nilpotent Lie groups. In this setting we completely characterize the long-time behaviour of the flow, showing that normalized solutions to the flow subconverge to a non-flat algebraic soliton, in Cheeger–Gromov topology. We also exhibit a uniqueness result for algebraic solitons on such Lie groups.

Ancient solutions for Andrews’ hypersurface flow

Abstract We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994. As time t → 0 - {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as t → - ∞ {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions S J × ℝ n - J {S^{J}\times\mathbb{R}^{n-J}} , 1 ≤ J ≤ n - 1 {1\leq J\leq n-1} . These results are the analog of the corresponding results in Ricci flow ( J = n - 1 {J=n-1} ) and mean curvature flow.

Hermitian curvature flow on unimodular Lie groups and static invariant metrics

We investigate the Hermitian curvature flow (HCF) of leftinvariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation ∂ t g t = − R i c 1 , 1 ( g t ) \partial _tg_{t}=-{\mathrm { Ric}}^{1,1}(g_t) . The solution g t g_t always exists for all positive times, and ( 1 + t ) − 1 g t (1 + t)^{-1}g_t converges as t → ∞ t\to \infty in Cheeger–Gromov sense to a nonflat left-invariant soliton ( G ¯ , g ¯ ) (\bar G, \bar g) . Moreover, up to homotheties on each of these groups there exists at most one left-invariant soliton solution, which is a static Hermitian metric if and only if the group is semisimple. In particular, compact quotients of complex semisimple Lie groups yield examples of compact non-Kähler manifolds with static Hermitian metrics. We also investigate the existence of static metrics on nilpotent Lie groups and we generalize a result of Enrietti, Fino, and the third author [J. Symplectic Geom. 10 (2012), no. 2, 203–223] for the pluriclosed flow. In the last part of the paper we study the HCF on Lie groups with abelian complex structures.

Boundary Harmonic Coordinates on Manifolds with Boundary in Low Regularity

In this paper, we prove the existence of $$H^2$$-regular coordinates on Riemannian 3-manifolds with boundary, assuming only $$L^2$$-bounds on the Ricci curvature, $$L^4$$-bounds on the second fundamental form of the boundary, and a positive lower bound on the volume radius. The proof follows by extending the theory of Cheeger–Gromov convergence to include manifolds with boundary in the above low regularity setting. The main tools are boundary harmonic coordinates together with elliptic estimates and a geometric trace estimate, and a rigidity argument using manifold doubling. Assuming higher regularity of the Ricci curvature, we also prove corresponding higher regularity estimates for the coordinates.

Eta and rho invariants on manifolds with edges

We establish existence of eta-invariants as well as of the Atiyah–Patodi–Singer and the Cheeger–Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature and the spin Dirac operator. We derive an analogue of the Atiyah–Patodi–Singer index theorem for incomplete edge spaces and their non-compact infinite Galois coverings with edge singular boundary. Our arguments are based on the microlocal analysis of the heat kernel asymptotics associated to the Dirac laplacian of an incomplete edge metric. As an application, we discuss stability results for the two rho-invariants we have defined.

Boundedness of spectral multipliers of generalized Laplacians on compact manifolds with boundary

Consider a second order, strongly elliptic negative semidefinite differential operator L (may be a system) on a compact Riemannian manifold overline{M} with smooth boundary, where the domain of L is defined by a coercive boundary condition. Classically known results, and also recent work in Duong et al. (J Funct Anal 196:443–485, 2002) and Duong and McIntosh (Rev Math Iberoam 15:233–265, 1999) establish sufficient conditions for L^infty -text {BMO}_L continuity of varphi (sqrt{-L}), where varphi in S^0_1({mathbb {R}}), and L is a suitable elliptic operator. Using a variant of the Cheeger–Gromov–Taylor functional calculus due to Mauceri et al. (Math Res Lett 16:861–879, 2009), and short time upper bounds on the integral kernel of e^{tL} due to Greiner (Arch Ration Mech Anal 41:168–218, 1971), we prove that a variant of such sufficient conditions holds for our operator L.

Knots having the same Seifert form and primary decomposition of knot concordance

We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance group and not concordant to any knot with coprime Alexander polynomial. Key ingredients for the proof are Cheeger–Gromov–von Neumann [Formula: see text]-invariants for amenable groups developed by Cha–Orr and polynomial splittings of metabelian [Formula: see text]-invariants.

Lower semicontinuity of mass under C0{C^{0}} convergence and Huisken’s isoperimetric mass

Abstract Given a sequence of asymptotically flat 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed C 0 {C^{0}} Cheeger–Gromov sense to an asymptotically flat limit space, we show that the total mass of the limit is bounded above by the liminf of the total masses of the sequence. In other words, total mass is lower semicontinuous under such convergence. In order to prove this, we use Huisken’s isoperimetric mass concept, together with a modified weak mean curvature flow argument. We include a brief discussion of Huisken’s work before explaining our extension of that work. The results are all specific to three dimensions.

A Myers-type compactness theorem by the use of Bakry–Emery Ricci tensor

Let (M,g) be a complete and connected Riemannian manifold of dimension n≥2. By using the Bakry–Emery Ricci curvature tensor on M, we prove a Myers-type compactness theorem which corresponds to the compactness theorem proved by Cheeger–Gromov–Taylor.

Some examples of vanishing Yamabe invariant and minimal volume, and collapsing of inequivalent smoothings and PL-structures

In this short note, exploits of constructions of \(\mathcal {F}\)-structures coupled with technology developed by Cheeger–Gromov and Paternain–Petean are seen to yield a procedure to compute minimal entropy, minimal volume, Yamabe invariant and to study collapsing with bounded sectional curvature on inequivalent smooth structures and inequivalent PL-structures within a fixed homeomorphism class. We compute these fundamental Riemannian invariants for every high-dimensional smooth manifold on the homeomorphism class of any smooth manifold that admits a Riemannian metric of zero sectional curvature. This includes all exotic and all fake tori of dimension greater than four. We observe that the minimal volume is not an invariant of the smooth structures, yet the Yamabe invariant does discern the standard smooth structure from all the others. We also observe that the fundamental group places no restriction on the vanishing of the minimal volume and collapse with bounded sectional curvature for high-dimensional manifolds.

Complexity of Surgery Manifolds and Cheeger–Gromov Invariants

We present new lower bounds on the complexity of Dehn surgery manifolds of knots, using our recent result on the Cheeger-Gromov rho invariants and triangulations. As an application, we give explicit examples of closed hyperbolic 3-manifolds with fixed first homology for which the gap between the Gromov norm and the complexity is arbitrarily large.

The Higson–Roe exact sequence and [formula omitted] eta invariants

The goal of this paper is to solve the problem of existence of an ℓ2 relative eta morphism on the Higson–Roe structure group. Using the Cheeger–Gromov ℓ2 eta invariant, we construct a group morphism from the Higson–Roe maximal structure group constructed in [35] to the reals. When we apply this morphism to the structure class associated with the spin Dirac operator for a metric of positive scalar curvature, we get the spin ℓ2 rho invariant. When we apply this morphism to the structure class associated with an oriented homotopy equivalence, we get the difference of the ℓ2 rho invariants of the corresponding signature operators. We thus get new proofs for the classical ℓ2 rigidity theorems of Keswani obtained in [41].