Injectivity Radius Research Articles

First eigenvalue of the Laplacian of a geodesic ball and area-based symmetrization of its metric tensor

Given a Riemmanian manifold, we provide a new method to compute a sharp upper bound for the first eigenvalue of the Laplacian for the Dirichlet problem on a geodesic ball of radius less than the injectivity radius of the manifold. This upper bound is obtained by transforming the metric tensor into a rotationally symmetric metric tensor that preserves the area of the geodesic spheres. The provided upper bound can be computed using only the area function of the geodesic spheres contained in the geodesic ball and it is sharp in the sense that the first eigenvalue of geodesic ball coincides with our upper bound if and only if the mean curvature pointed inward of each geodesic sphere is a radial function.

Foliations on closed three-dimensional Riemannian manifolds with small modulus of mean curvature of the leaves

We prove that the modulus of mean curvature of the leaves of a transversely oriented foliation of codimension one with a generalized Reeb component on an oriented smooth closed three-dimensional Riemannian manifold cannot be everywhere smaller than a certain positive constant depending on the volume, the maximum value of the sectional curvature, and the injectivity radius of the manifold. This means that foliations with small modulus of mean curvature of the leaves are taut.

Fridman Function, Injectivity Radius Function, and Squeezing Function

Very recently, the Fridman function of a complex manifold X has been identified as a dual of the squeezing function of X. In this paper, we prove that the Fridman function for certain hyperbolic complex manifold X is bounded above by the injectivity radius function of X. This result also suggests us to use the Fridman function to extend the definition of uniform thickness to higher dimensional hyperbolic complex manifolds. We also establish an expression for the Fridman function (with respect to the Kobayashi metric) when $$X = \mathbb {D} \diagup \varGamma $$ and $$\varGamma $$ is a torsion-free discrete subgroup of isometries on the standard open unit disk $$\mathbb {D}$$ . Hence explicit formulae of the Fridman functions for the annulus $$A_r$$ and the punctured disk $$\mathbb {D}^*$$ are derived. These are the first explicit non-constant Fridman functions. Finally, we explore the boundary behavior of the Fridman functions (with respect to the Kobayashi metric) and the squeezing functions for regular type hyperbolic Riemann surfaces and planar domains, respectively.

Finite topological type of complete Finsler gradient shrinking Ricci solitons

In the present work it is shown that on a Finslerian space, a forward complete gradient shrinking Ricci soliton has finite topological type, provided either the Ricci scalar is bounded above or the Ricci scalar is bounded from below and injectivity radius is bounded away from zero.

Maximal characterisation of local Hardy spaces on locally doubling manifolds

We prove a radial maximal function characterisation of the local atomic Hardy space {{mathfrak {h}}}^1(M) on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to {{mathfrak {h}}}^1(M) if and only if either its local heat maximal function or its local Poisson maximal function is integrable. A key ingredient is a decomposition of Hölder cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A. Uchiyama.

Upper bounds for Steklov eigenvalues of submanifolds in Euclidean space via the intersection index

We obtain upper bounds for the Steklov eigenvalues σk(M) of a smooth, compact, n-dimensional submanifold M of Euclidean space with boundary Σ that involve the intersection indices of M and of Σ. One of our main results is an explicit upper bound in terms of the intersection index of Σ, the volume of Σ and the volume of M as well as dimensional constants. By also taking the injectivity radius of Σ into account, we obtain an upper bound that has the optimal exponent of k with respect to the asymptotics of the Steklov eigenvalues as k→∞.

Critical and subcritical Trudinger-Moser inequalities on complete noncompact Riemannian manifolds

Though the sharp Trudinger-Moser inequalities on compact Riemannian manifolds have been known for some times, optimal constants for such inequalities on complete noncompact Riemannian manifolds are still unknown except in some special cases (such as hyperbolic spaces or Hadamard manifolds). In this paper, we will establish the sharp critical and subcritical Trudinger-Moser type inequalities with best constants on any complete and noncompact Riemannian manifold (M,g) whose Ricci curvature and its injectivity radius have lower bounds. (See Theorems 1.3 and 1.4.)More precisely, we will establish the following sharp critical Trudinger-Moser inequality on such manifolds (M,g,dVg):supu∈W1,n(M),||u||1,τ≤1⁡∫Mϕ(αn|u|nn−1)dVg≤C(n,τ), where τ>0, ϕ(t)=∑k=n−1∞tkk!, αn=nωn−11n−1, and ωn−1 is the area of the unit sphere in Rn, ||u||1,τ=(∫Mτ|u|n+|∇u|ndVg)1n. Moreover, we also prove that the following sharp subcritical Trudinger-Moser inequalitysupu∈W1,n(M),||∇u||n≤1⁡1||u||Ln(M)n∫Mϕ(α|u|nn−1)dVg≤C(n,α) in the sense that the supremum is finite for all α<αn, but infinite for α≥αn.The proofs of both the critical and subcritical Trudinger-Moser inequalities rely crucially on the Trudinger-Moser inequality on any compact manifold M with or without boundary, which is of its own interest. The important property we establish in this paper is that the upper bound C(n,M) for the Trudinger-Moser supremum is continuous and monotone increasing with respect to the volume of M. (See Theorems 1.1 and 1.2.) This upper bound property allows us to carry out a symmetrization-free argument from sharp local inequalities on compact manifolds to the optimal global inequalities on complete and noncompact Riemannian manifolds. Our optimal results sharpen substantially the existing ones on such Riemannian manifolds in the literature.

A volume comparison theorem for characteristic numbers

We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of certain characteristic numbers of a Riemannian manifold, including all Pontryagin and Chern numbers, is bounded proportionally to the volume. The proof relies on Chern–Weil theory applied to a connection constructed from Euclidean connections on charts in which the metric tensor is harmonic and has bounded Hölder norm. We generalize this theorem to a Gromov–Hausdorff closed class of rough Riemannian manifolds defined in terms of Hölder regularity. Assuming an additional upper Ricci curvature bound, we show that also the Euler characteristic is bounded proportionally to the volume. Additionally, we remark on a volume comparison theorem for Betti numbers of manifolds with an additional upper bound on sectional curvature. It is a consequence of a result by Bowen.

On the Free Surface Motion of Highly Subsonic Heat-Conducting Inviscid Flows

For the free surface problem of highly subsonic heat-conducting inviscid flow in 2-D and 3-D, a priori estimates for geometric quantities of free surfaces, such as the second fundamental form and the injectivity radius of the normal exponential map, and the Sobolev norms of fluid variables, are proved by investigating the coupling of the boundary geometry and the interior solutions. An interesting feature for the free surface problem studied in this paper is the loss of one more derivative than the problem of incompressible Euler equations, for which a geometric approach was introduced by Christodoulou and Lindblad [11]. Due to the loss of the one more derivative and loss of the symmetry of equations which create significant difficulties in closing the estimates in Sobolev spaces of finite regularity, the geometric approach in [11] needs to be substantially developed and extended by exploring the interaction of large variation of temperature, heat-conduction, non-zero divergence of the fluid velocity and the evolution of free surfaces.

Short geodesic loops and $$L^p$$ norms of eigenfunctions on large genus random surfaces

We give upper bounds for $$L^p$$ norms of eigenfunctions of the Laplacian on compact hyperbolic surfaces in terms of a parameter depending on the growth rate of the number of short geodesic loops passing through a point. When the genus $$g \rightarrow +\infty $$ , we show that random hyperbolic surfaces X with respect to the Weil-Petersson volume have with high probability at most one such loop of length less than $$c \log g$$ for small enough $$c > 0$$ . This allows us to deduce that the $$L^p$$ norms of $$L^2$$ normalised eigenfunctions on X are $$O(1/\sqrt{\log g})$$ with high probability in the large genus limit for any $$p > 2 + \varepsilon $$ for $$\varepsilon > 0$$ depending on the spectral gap $$\lambda _1(X)$$ of X, with an implied constant depending on the eigenvalue and the injectivity radius.

On the boundary injectivity radius of Buser–Colbois–Dodziuk-Margulis tubes

We give a lower bound on the boundary injectivity radius of the Margulis tubes with smooth boundary constructed by Buser, Colbois, and Dodziuk. This estimate depends on the dimension and a curvature bound only.

Uniform bounds on harmonic Beltrami differentials and Weil–Petersson curvatures

Abstract In this article we show that for every finite area hyperbolic surface X of type {(g,n)} and any harmonic Beltrami differential μ on X, then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of μ over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to {-g} as the genus g goes to infinity.

Estimates of the Bergman kernel on a hyperbolic Riemann surface of finite volume-II

Dans cet article, nous dérivons des estimations non-diagonales du noyau de Bergman associé aux puissances tensorielles du faisceau cotangent défini sur une surface de Riemann hyperbolique de volume fini, lorsque la distance entre les points est inférieure au rayon d’injectivité. Nous utilisons ensuite ces estimations pour dériver des estimations du noyau de Bergman le long de la diagonale.

Quasihyperbolic metric universal covers

We prove that the natural metric universal cover of a quasihyperbolic plane domain is a Hadamard metric space with injectivity radius \pi . This in turn provides sharp constants for several conjectures by Väisälä.

Examples of compact Einstein four-manifolds with negative curvature

We give new examples of compact, negatively curved Einstein manifolds of dimension 4 4 . These are seemingly the first such examples which are not locally homogeneous. Our metrics are carried by a sequence of four-manifolds ( X k ) (X_k) previously considered by Gromov and Thurston (Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), no. 1, 1–12). The construction begins with a certain sequence ( M k ) (M_k) of hyperbolic four-manifolds, each containing a totally geodesic surface Σ k \Sigma _k which is nullhomologous and whose normal injectivity radius tends to infinity with k k . For a fixed choice of natural number l l , we consider the l l -fold cover X k → M k X_k \to M_k branched along Σ k \Sigma _k . We prove that for any choice of l l and all large enough k k (depending on l l ), X k X_k carries an Einstein metric of negative sectional curvature. The first step in the proof is to find an approximate Einstein metric on X k X_k , which is done by interpolating between a model Einstein metric near the branch locus and the pull-back of the hyperbolic metric from M k M_k . The second step in the proof is to perturb this to a genuine solution to Einstein’s equations, by a parameter dependent version of the inverse function theorem. The analysis relies on a delicate bootstrap procedure based on L 2 L^2 coercivity estimates.

Lower bounds for the volume with upper bounds for the Ricci Curvature in dimension three

In this note we provide several lower bounds for the volume of a geodesic ball within the injectivity radius in a $3$-dimensional Riemannian manifold assuming only upper bounds for the Ricci curvature.

Energy distribution of harmonic 1-forms and Jacobians of Riemann surfaces with a short closed geodesic

We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface S where a short closed geodesic is pinched. If the geodesic separates the surface into two parts, then the Jacobian variety of S develops into a variety that splits. If the geodesic is nonseparating then the Jacobian degenerates. The aim of this work is to get insight into this process and give estimates in terms of geometric data of both the initial surface S and the final surface, such as its injectivity radius and the lengths of geodesics that form a homology basis. The Jacobians in this paper are represented by Gram period matrices. As an invariant we introduce new families of symplectic matrices that compensate for the lack of full dimensional Gram-period matrices in the noncompact case.

Scattering theory without injectivity radius assumptions, and spectral stability for the Ricci flow

We prove a completely new integral criterion for the existence and completeness of the wave operators $W_{\pm}(-\Delta_h,-\Delta_g, I_{g,h})$ corresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami operators $-\Delta_j$, $j=1,2$, that are induced by two quasi-isometric complete Riemannian metrics $g$ and $h$ on an open manifold $M$. In particular, this result provides a criterion for the absolutely continuous spectra of $-\Delta_g$ and $-\Delta_h$ to coincide. Our proof relies on estimates that are obtained using a probabilistic Bismut type formula for the gradient of a heat semigroup. Unlike all previous results, our integral criterion only requires some lower control on the Ricci curvatures and some upper control on the heat kernels, but no control at all on the injectivity radii. As a consequence, we obtain a stability result for the absolutely continuous spectrum under a Ricci flow.

The maximal injectivity radius of hyperbolic surfaces with geodesic boundary

We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author's result to the with-boundary setting. In the second part of the paper we comment on another direction for extending this result, via the systole of loops function.

Isoperimetric Inequalities in Riemann Surfaces and Graphs

A celebrated theorem of Kanai states that quasi-isometries preserve isoperimetric inequalities between uniform Riemannian manifolds (with positive injectivity radius) and graphs. Our main result states that we can study the (Cheeger) isoperimetric inequality in a Riemann surface by using a graph related to it, even if the surface has injectivity radius zero (this graph is inspired in Kanai’s graph, but it is different from it). We also present an application relating Gromov boundary and isoperimetric inequality.