This work proves certain general orbifold compactness results for spaces of Riemannian metrics, generalizing earlier results along these lines for Einstein metrics or metrics with bounded Ricci curvature. This is then applied to prove such compactness for spaces of Bach-flat (for example half-conformally flat) metrics on 4-manifolds, and related results for metrics which are critical points of other natural Riemannian functionals on the space of metrics.
A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold which is locally modeled on the quotient of a connected, open manifold under a finite group of isometries. If all of the isometries used to define the local structures of an entire orbifold are orientation preserving, we call the orbifold locally orientable. We use heat invariants to show that a Riemannian orbifold which is locally orientable cannot be Laplace isospectral to a Riemannian orbifold which is not locally orientable. As a corollary we observe that a Riemannian orbifold that is not locally orientable …
We classify positively curved Alexandrov spaces of dimension 4 with an isometric circle action up to equivariant homeomorphism, subject to a certain additional condition on the infinitesimal geometry near fixed points which we conjecture is always satisfied. As a corollary, we also classify positively curved Riemannian orbifolds of dimension 4 with an isometric circle action.
We study the geometry and topology of Riemannian 3-orbifolds which are locally volume collapsed with respect to a curvature scale. We show that a sufficiently collapsed closed 3-orbifold without bad 2-suborbifolds either admits a metric of nonnegative sectional curvature or satisfies Thurston's Geometrization Conjecture. We also prove a version with boundary. Kleiner and Lott independently proved similar results.
We characterize Riemannian orbifolds and their coverings in terms of metric geometry. In particular, we show that the metric double of a Riemannian orbifold along the closure of its codimension one stratum is a Riemannian orbifold and that the natural projection is an orbifold covering.
Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces which have nonnegative generalized scalar curvature and Euclidean tangent cones almost everywhere. In this paper we prove this conjecture for sequences of rotationally symmetric warped product manifolds. We show that the limit spaces have $H^1$ warping function that has nonnegative scalar curvature in a weak sense, and have Euclidean tangent cones almost ever…
In this paper we study the local regularity of closed surfaces immersed in a Riemannian 3-manifold flowing by Willmore flow. We establish a pair of concentration-compactness alternatives for the flow, giving a lower bound on the maximal time of existence of the flow proportional to the concentration of the curvature and area at initial time. The estimate from the first theorem is purely in terms of the concentration of curvature at initial time but applies only to ambient spaces with non-positive sectional curvature. The second requires additional information on the concentration of area at in…
We characterize Riemannian orbifolds with an upper curvature bound in the Alexandrov sense as reflectofolds, i.e. Riemannian orbifolds all of whose local groups are generated by reflections, with the same upper bound on the sectional curvature. Combined with a result by Lytchak--Thorbergsson this implies that a quotient of a Riemannian manifold by a closed group of isometries has locally bounded curvature (from above) in the Alexandrov sense if and only if it is a reflectofold.
Let (M^n_i,g_i,p_i) be a sequence of smooth pointed complete n-dimensional Riemannian Manifolds with uniform bounds on the sectional curvatures and let (X,d,p) be a metric space such that (M^n_i,g_i,p_i) -> (X,d,p) in the Gromov-Hausdorff sense. Let O \subseteq X be the set of points x \in X such that there exists a neighborhood of x which is isometric to an open set in a Riemannian orbifold and let B = O^c be the complement set. Then we have the sharp estimates dim_Haus(B) \leq min{n-5, dim_Haus(X)-3}, and further for arbitrary x \in X we have that x \in O iff a neighborhood of x has bounded …
We prove a formula for the normal injectivity radius(thickness)i(K,M)for C^{1,1} compact submanifolds K^k of complete Riemannian manifolds M^n in terms of geometric focal distance and double critical points. We also prove the C^1 compactness of the set of all compact submanifolds K contained in a compact subset D of a fixed manifold M with i(K,M) bounded away from 0. This is an extrinsic and isometric embeddindg type theorem related to Gromov's compactness theorem. This yields the existence of thickest submanifolds in an isotopy class where K is not necessarily connected.
In this paper we study the asymptotic behavior of second-order uniformly elliptic operators on weighted Riemannian manifolds. They naturally emerge when studying spectral properties of the Laplace-Beltrami operator on families of manifolds with rapidly oscillating metrics. We appeal to the notion of H-convergence introduced by Murat and Tartar. In our main result we establish an H-compactness result that applies to elliptic operators with measurable, uniformly elliptic coefficients on weighted Riemannian manifolds. We further discuss the special case of ``locally periodic'' coefficients and st…
In arXiv:1005.3255 we proved an orbifold Cheeger-Gromov compactness theorem for complete 4d Ricci shrinkers with a lower bound for the entropy, an upper bound for the Euler characterisic, and a lower bound for the gradient of the potential at large distances. In this note, we show that the last two assumptions in fact can be removed. The key ingredient is a recent estimate of Cheeger-Naber arXiv:1406.6534.
Given a (possibly singular) Riemannian foliation $\mathcal{F}$ with closed leaves on a compact manifold $M$ with an adapted metric, we investigate the wave trace invariants for the basic Laplacian about a non-zero period. We compare them to the wave invariants of the underlying Riemannian orbifold that exists when the leaves in the regular region are identified to points, equipped with the metric that is transverse to the leaves of the foliation. Recalling that the basic Laplacian differs from the underlying orbifold Laplacian by a term that is the mean curvature vector field associated to the…
The class of Riemannian orbifolds of dimension n defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphism. Furthermore, any class of isospectral Riemannian orbifolds with a lower bound on the sectional curvature is finite up to orbifold homeomorphism.
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…
We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss-Bonnet with cutoff argument.
An old problem asks whether a Riemannian manifold can be isospectral to a Riemannian orbifold with nontrivial singular set. In this short note we show that under the assumption of Schanuel's conjecture in transcendental number theory, this is impossible whenever the orbifold and manifold in question are length-commensurable compact locally symmetric spaces of nonpositive curvature associated to simple Lie groups.
We introduce the Γ-extension of the spectrum of the Laplacian of a Riemannian orbifold, where Γis a finitely generated discrete group. This extension, called the Γ-spectrum, is the union of the Laplace spectra of the Γ-sectors of the orbifold, and hence constitutes a Riemannian invariant that is directly related to the singular set of the orbifold. We compare the Γ-spectra of known examples of isospectral pairs and families of orbifolds and demonstrate that it many cases, isospectral orbifolds need not be Γ-isospectral. We additionally prove a version of Sunada's theorem that allows us to cons…
We prove that the quotient space of a variationally complete group action is a good Riemannian orbifold. The result is generalized to singular Riemannian foliations without horizontal conjugate points.
The regularity of limit spaces of Riemannian manifolds with L^p curvature bounds, $p > n/2$, is investigated under no apriori non-collapsing assumption. A regular subset, defined by a local volume growth condition for a limit measure, is shown to carry the structure of a Riemannian manifold. One consequence of this is a compactness theorem for Riemannian manifolds with $L^p$ curvature bounds and an a priori volume growth assumption in the pointed Cheeger-Gromov topology. A different notion of convergence is also studied, which replaces the exhaustion by balls in the pointed Cheeger-Gromov topo…
In recent years a lot of attention has been paid to topological spaces which are a bit more general than smooth manifolds - orbifolds. Orbifolds are intuitively speaking manifolds with some singularities. The formal definition is also modelled on that of manifolds, an orbifold is a topological space which locally is homeomorphic to the orbit space of a finite group acting on $R^n$. Orbifolds were defined by Satake, as V-manifolds, then studied by W. Thurston, who introduced the term "orbifold". Due to their importance in physics, and in particular in the string theory, orbifolds have been draw…
We study the relationship between the geometry and the Laplace spectrum of a Riemannian orbifold O via its heat kernel; as in the manifold case, the time-zero asymptotic expansion of the heat kernel furnishes geometric information about O. In the case of a good Riemannian orbifold (i.e., an orbifold arising as the orbit space of a manifold under the action of a discrete group of isometries), H. Donnelly proved the existence of the heat kernel and constructed the asymptotic expansion for the heat trace. We extend Donnelly's work to the case of general compact orbifolds. Moreover, in both the go…
Let a compact Lie group act isometrically on a non-collapsing sequence of compact Alexandrov spaces with fixed dimension and uniform lower curvature and upper diameter bounds. If the sequence of actions is equicontinuous and converges in the equivariant Gromov--Hausdorff topology, then the limit space is equivariantly homeomorphic to spaces in the tail of the sequence. As a consequence, the class of Riemannian orbifolds of dimension $n$ defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphi…
In this paper we survey on some recent results on Riemannian orbifolds and singular Riemannian foliations and combine them to conclude the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.f. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.
We prove the compactness of solutions to general fourth order elliptic equations which are L^1-perturbations of the Q-curvature equation on compact Riemannian 4-maniods. Consequently, we prove the global existence and convergence of the Q-curvature flow on a generic class of Riemannian 4-manifolds. As a by product, we give a positive answer to an open question by A. Malchiodi on the existence of bounded Palais-Smale sequences for the Q-curvature problem when the Paneitz operator is positive with trivial kernel.
We show that suitably defined systolic ratios are globally bounded from above on the space of rotationally symmetric spindle orbifolds and that the upper bound is attained precisely at so-called Besse metrics, i.e. Riemannian orbifold metrics all of whose geodesics are closed.
Metrics on Lie groupoids and differentiable stacks have been introduced recently, extending the Riemannian geometry of manifolds and orbifolds to more general singular spaces. Here we continue that theory, studying stacky curves on Riemannian stacks, measuring their length using stacky metrics, and introducing stacky geodesics. Our main results show that the length of stacky curves measure distances on the orbit space, characterize stacky geodesics as locally minimizing curves, and establish a stacky version of Hopf-Rinow Theorem. We include a concise overview that bypasses nonessential techni…
We prove a compactness theorem for metrics with Bounded Integral Curvature on a fixed closed surface $Σ$. As a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities, where an accumulation of singularities is allowed.
Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the stability operator. When the latter assumption is replaced by a uniform lower bound on the p-th Jacobi eigenvalue for p greater or equal than 2 one gains strong convergence to a smooth limit submanifold away from at most p-1 points.
We show that if $g$ is a Riemannian metric on a closed piecewise locally symmetric manifold $M$, then the lift of $g$ to the universal cover $\widetilde{M}$ has a discrete isometry group. We also show that the index $[\Isom(\widetilde{M}): π_1(M)]$ is bounded by a constant independent of $g$.
Weak Continuity of the Cartan Structural System and Compensated Compactness on Semi-Riemannian Manifolds with Lower Regularity
Mathematical Institute, University of Oxford（牛津大学数学研究所）；New York University–Shanghai（纽约大学上海分校）；NYU–ECNU Institute of Mathematical Sciences（纽约大学-华东师范大学数学科学研究所）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
In this paper, we try to generalize to the case of compact Riemannian orbifolds $Q$ some classical results about the existence of closed geodesics of positive length on compact Riemannian manifolds $M$. We shall also consider the problem of the existence of infinitely many geometrically distinct closed geodesics. In the classical case the solution of those problems involve the consideration of the homotopy groups of $M$ and the homology properties of the free loop space on $M$(Morse theory). Those notions have their analogue in the case of orbifolds (see [7]). The main part of this paper will …
Let $(M, g)$ be a closed Riemannian manifold of dimension $5$. Assume that $(M, g)$ is not conformally equivalent to the round sphere. If the scalar curvature $R_g\geq 0$ and the $Q$-curvature $Q_g\geq 0$ on $M$ with $Q_g(p)>0$ for some point $p\in M$, we prove that the set of metrics in the conformal class of $g$ with prescribed constant positive $Q$-curvature is compact in $C^{4, α}$ for any $0 <α< 1$. We also give some estimates for dimension $6$ and $7$.
We construct bi-Lipschitz embeddings into Euclidean space for manifolds and orbifolds of bounded diameter and curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. Our results also apply for bounded subsets of complete Riemannian manifolds, and complete flat and elliptic orbifolds. Our approach is based on analysing the structure of a bounded curvature manifold at various scales by specializing methods from collapsing theory to a certain class of model spaces.
We consider the $G$-invariant spectrum of the Laplacian on an orbit space $M/G$ where $M$ is a compact Riemannian manifold and $G$ acts by isometries. We generalize the Sunada-Pesce-Sutton technique to the $G$-invariant setting to produce pairs of isospectral non-isometric orbit spaces. One of these spaces is isometric to an orbifold with constant sectional curvature whereas the other admits non-orbifold singularities and therefore has unbounded sectional curvature. We therefore show that constant sectional curvature and the presence of non-orbifold singularities are inaudible to the G-invaria…
Let $\mathcal{K}(n, V)$ be the set of $n$-dimensional compact Kahler-Einstein manifolds $(X, g)$ satisfying $Ric(g)= - g$ with volume bounded above by $V$. We prove that after passing to a subsequence, any sequence $\{ (X_j, g_j)\}_{j=1}^\infty$ in $\mathcal{K}(n, V)$ converges, in the pointed Gromov-Hausdorff topology, to a finite union of complete Kahler-Einstein metric spaces without loss of volume. The convergence is smooth off a closed singular set of Hausdorff dimension no greater than $2n-4$, and the limiting metric space is biholomorphic to an $n$-dimensional semi-log canonical model w…
The theory of twistors on foliated manifolds is developed and the twistor space of the normal bundle is constructed. It is demonstrated that the classical constructions of the twistor theory lead to foliated objects and permit to formulate and prove foliated versions of some well-known results on holomorphic mappings. Since any orbifold can be understood as the leaf space of a suitable defined Riemannian foliation we obtain orbifold versions of the classical results as a simple consequence of the results on foliated mappings.
We consider the moduli space of the extremal Kähler metrics on compact manifolds. We show that under the conditions of two-sided total volume bounds, $L^{n\over2}$-norm bounds on $\Riem$, and Sobolev constant bounds, this Moduli space can be compactified by including (reduced) orbifolds with finitely many singularities. Most of our results go through for certain other classes of critical Riemannian metrics.
