Convergence Of Manifolds Research Articles

Convergence of manifolds and metric spaces with boundary

We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. We prove theorems demonstrating when the Gromov–Hausdorff (GH) and Sormani–Wenger Intrinsic Flat (SWIF) limits of sequences of such metric spaces agree. Thus in particular the limit spaces are countably [Formula: see text] rectifiable spaces. From these theorems we derive compactness theorems for sequences of Riemannian manifolds with boundary where both the GH and SWIF limits agree. For sequences of Riemannian manifolds with boundary we only require non-negative Ricci curvature, upper bounds on volume, noncollapsing conditions on the interior of the manifold and diameter controls on the level sets near the boundary.

Intrinsic Flat and Gromov-Hausdorff Convergence of Manifolds with Ricci Curvature Bounded Below

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature bounded below and diameter bounded above, Gromov-Hausdorff convergence agrees with intrinsic flat convergence. In particular, the limiting current is essentially unique, has multiplicity one, and mass equal to the Hausdorff measure. Moreover, the limit spaces satisfy a constancy theorem.

A universal Riemannian foliated space

It is proved that the isometry classes of pointed connected complete Riemannian n-manifolds form a Polish space, M⁎∞(n), with the topology described by the C∞ convergence of manifolds. This space has a canonical partition into sets defined by varying the distinguished point into each manifold. The locally non-periodic manifolds define an open dense subspace M⁎,lnp∞(n)⊂M⁎∞(n), which becomes a C∞ foliated space with the restriction of the canonical partition. Its leaves without holonomy form the subspace M⁎,np∞(n)⊂M⁎,lnp∞(n) defined by the non-periodic manifolds. Moreover the leaves have a natural Riemannian structure so that M⁎,lnp∞(n) becomes a Riemannian foliated space, which is universal among all sequential Riemannian foliated spaces satisfying certain property called covering-determination. M⁎,lnp∞(n) is used to characterize the realization of complete connected Riemannian manifolds as dense leaves of covering-determined compact sequential Riemannian foliated spaces.

Erratum to: On eigenvalue pinching in positive curvature

Since volM ≤ volSn this does not affect the arguments below. The other more serious mistake is that the set Aδ need not be connected so we cannot assume that det (dΦ) has constant sign on this set. This means that the proof of Lemma 6.2 is incomplete as it stands. The author does not see a way of correcting this particular error nor does he know of an alternate proof which is simpler than what is already in the literature. The author would like to mention that Cheeger has written an article [1], which contains a complete and streamlined account on convergence of manifolds with Ricci curvature bounds. Note that this does not affect Theorem 1.1 of [2]. This result is proven in Sects. 3, 4, 5 and does not depend on Sect. 6.

Congruences and quasiidentities on unars

We establish conditions under which the fact that all congruences of two unars (universal algebras with one unary operation) are convergent implies the convergence of manifolds and quasimanifolds generated by these unars.

Lipschitz convergence of manifolds of positive Ricci curvature with large volume

Lipschitz convergence of manifolds of positive Ricci curvature with large volume