Pucci eigenvalues on geodesic balls

We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng’s eigenvalue comparison theorem for the Laplace–Beltrami operator. For manifolds with bounded sectional curvature, we prove Cheng’s bounds hold for Pucci eigenvalues on geodesic balls of radius less than the injectivity radius. For manifolds with Ricci curvature bounded below, we prove Cheng’s upper bound holds for Pucci eigenvalues on certain small geodesic balls. We also prove that the principal Pucci eigenvalues of an \({O(n)}\)-invariant hypersurface immersed in \({{\mathbb{R}}^{n+1}}\) with one smooth boundary component are smaller than the eigenvalues of an \({n}\)-dimensional Euclidean ball with the same boundary.     

Upper Bounds on the First Eigenvalue for a Diffusion Operator via Bakry–Émery Ricci Curvature II

Let ${L=\Delta-\nabla\varphi\cdot\nabla}$ be a symmetric diffusion operator with an invariant measure ${d\mu=e^{-\varphi}dx}$ on a complete Riemannian manifold. In this paper we prove Li–Yau gradient estimates for weighted elliptic equations on the complete manifold with ${|\nabla \varphi| \leq \theta}$ and ∞-dimensional Bakry–Émery Ricci curvature bounded below by some negative constant. Based on this, we give an upper bound on the first eigenvalue of the diffusion operator L on this kind manifold, and thereby generalize a Cheng’s result on the Laplacian case (Math Z, 143:289–297, 1975).     

Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound

We complete the picture of sharp eigenvalue estimates for the \(p\) -Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator \(\Delta _p\) when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces.     

Sharp local estimates for the Szegö–Weinberger profile in Riemannian manifolds

We study the local Szegö–Weinberger profile in a geodesic ball \(B_g(y_0,r_0)\) centered at a point \(y_0\) in a Riemannian manifold \(({\mathcal {M}},g)\) . This profile is obtained by maximizing the first nontrivial Neumann eigenvalue \(\mu _2\) of the Laplace–Beltrami Operator \(\Delta _g\) on \({\mathcal {M}}\) among subdomains of \(B_g(y_0,r_0)\) with fixed volume. We derive a sharp asymptotic bounds of this profile in terms of the scalar curvature of \({\mathcal {M}}\) at \(y_0\) . As a corollary, we deduce a local comparison principle depending only on the scalar curvature. Our study is related to previous results on the profile corresponding to the minimization of the first Dirichlet eigenvalue of \(\Delta _g\) , but additional difficulties arise due to the fact that \(\mu _2\) is degenerate in the unit ball in \(\mathbb {R}^N\) and geodesic balls do not yield the optimal lower bound in the asymptotics we obtain.     

Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diff _c (M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S ¹, the geodesic distance on Diff _c (S ¹) vanishes if and only if ${s\leq\frac12}$ . For other manifolds, we obtain a partial characterization: the geodesic distance on Diff _c (M) vanishes for ${M=\mathbb{R}\times N, s < \frac12}$ and for ${M=S^1\times N, s\leq\frac12}$ , with N being a compact Riemannian manifold. On the other hand, the geodesic distance on Diff _c (M) is positive for ${{\rm dim}(M)=1, s > \frac12}$ and dim(M) ≥ 2, s ≥ 1. For ${M=\mathbb{R}^n}$ , we discuss the geodesic equations for these metrics. For n = 1, we obtain some well-known PDEs of hydrodynamics: Burgers’ equation for s = 0, the modified Constantin–Lax–Majda equation for ${s=\frac12}$ , and the Camassa–Holm equation for s = 1.     

Positive solutions of quasilinear elliptic inequalities on noncompact Riemannian manifolds

In this paper, we consider the generalized solutions of the inequality $$ - div(A(x,u,\nabla u)\nabla u) \geqslant F(x,u,\nabla u)u^q , q > 1,$$ on noncompact Riemannian manifolds. We obtain sufficient conditions for the validity of Liouville’s theorem on the triviality of the positive solutions of the inequality under consideration. We also obtain sharp conditions for the existence of a positive solution of the inequality ? Δu ≥ u ^q, q > 1, on spherically symmetric noncompact Riemannian manifolds.     

Eigenvalue estimates for generalized Dirac operators on Sasakian manifolds

We consider a two-parameter generalization $D_{ab}$ of the Riemann Dirac operator $D$ on a closed Sasakian spin manifold, focusing attention on eigenvalue estimates for $D_{ab}$ . We investigate a Sasakian version of twistor spinors and Killing spinors, applying it to establish a new connection deformation technique that is adapted to fit with the Sasakian structure. Using the technique and the fact that there are two types of eigenspinors of $D_{ab}$ , we prove several eigenvalue estimates for $D_{ab}$ which improve Friedrich’s estimate (Friedrich, Math Nachr 97, 117–146, 1980).     

D’Atri Spaces of Type k and Related Classes of Geometries Concerning Jacobi Operators

In this article, we continue the study of the geometry of k-D’Atri spaces, 1≤k ≤n?1 (n denotes the dimension of the manifold), begun by the second author. It is known that k-D’Atri spaces, k≥1, are related to properties of Jacobi operators R _v along geodesics, since she has shown that ${\operatorname{tr}}R_{v}$ , ${\operatorname{tr}}R_{v}^{2}$ are invariant under the geodesic flow for any unit tangent vector v. Here, assuming that the Riemannian manifold is a D’Atri space, we prove in our main result that ${ \operatorname{tr}}R_{v}^{3}$ is also invariant under the geodesic flow if k≥3. In addition, other properties of Jacobi operators related to the Ledger conditions are obtained, and they are used to give applications to Iwasawa type spaces. In the class of D’Atri spaces of Iwasawa type, we show two different characterizations of the symmetric spaces of noncompact type: they are exactly the $\frak{C}$ -spaces, and on the other hand they are k -D’Atri spaces for some k≥3. In the last case, they are k-D’Atri for all k=1,…,n?1 as well. In particular, Damek–Ricci spaces that are k -D’Atri for some k≥3 are symmetric. Finally, we characterize k-D’Atri spaces for all k=1,…,n?1 as the $\frak{SC}$ -spaces (geodesic symmetries preserve the principal curvatures of small geodesic spheres). Moreover, applying this result in the case of 4 -dimensional homogeneous spaces, we prove that the properties of being a D’Atri (1-D’Atri) space, or a 3-D’Atri space, are equivalent to the property of being a k-D’Atri space for all k=1,2,3.     

The isoperimetric profile of a noncompact Riemannian manifold for small volumes

In the main theorem of this paper we treat the problem of existence of minimizers of the isoperimetric problem in a noncompact Riemannian manifold $M$ , under the assumption of small volumes. We use a new approach to the lack of compactness of this problem. Thanks to compactness theorems of the theory of the convergence of Riemannian manifolds we are able to prove, under suitable bounded geometry assumptions, that isoperimetric regions for small volumes always exist in a larger manifold obtained by attaching to the original one a pointed Gromov-Hausdorff limit of a sequence $(M,g,p_i)$ for a diverging sequence of points $p_i\in M$ . Applications of the main theorem to asymptotic expansions of the isoperimetric problem are given.     

First eigenvalues of geometric operators under the Yamabe flow

Suppose \((M,g_0)\) is a compact Riemannian manifold without boundary of dimension \(n\ge 3\). Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of \(g_0\) with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold.     

On the Perelman’s reduced entropy and Ricci flat manifolds with maximal volume growth

In this paper, we study the Ricci flat manifolds with maximal volume growth using Perelman’s reduced volume of Ricci flow. We show that if $(M^n,g)$ is an noncompact complete Ricci flat manifold with maximal volume growth satisfying $|Rm|(x)\rightarrow 0$ as $d(x)=d_g(x,p)\rightarrow \infty $ , then $M^n$ has the quadratic curvature decay. Some applications to this result are also presented.     

Generalized Killing spinors on spheres

We study generalized Killing spinors on round spheres \(\mathbb {S}^n\) . We show that on the standard sphere \(\mathbb {S}^8\) any generalized Killing spinor has to be an ordinary Killing spinor. Moreover, we classify generalized Killing spinors on \(\mathbb {S}^n\) whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola–Friedrich’s canonical spinor on 3-Sasakian manifolds of dimension 7. Finally, we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors.     

Global symplectic coordinates on gradient Kähler–Ricci solitons

A classical result of McDuff [14] asserts that a simply connected complete Kähler manifold $(M,g,\omega )$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi \ : M \rightarrow \mathbb{R }^{2n}$ (where $n$ is the complex dimension of $M$ ), satisfying the following property (proved by E. Ciriza in [4]): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi (p)=0$ , is a complex linear subspace of $\mathbb C ^n\simeq \mathbb{R }^{2n}$ . The aim of this paper is to exhibit, for all positive integers $n$ , examples of $n$ -dimensional complete Kähler manifolds with non-negative sectional curvature globally symplectomorphic to $\mathbb{R }^{2n}$ through a symplectomorphism satisfying Ciriza’s property.     

Robust vanishing of all Lyapunov exponents for iterated function systems

Given any compact connected manifold $M$ , we describe $C^2$ -open sets of iterated functions systems (IFS’s) admitting fully-supported ergodic measures whose Lyapunov exponents along $M$ are all zero. Moreover, these measures are approximated by measures supported on periodic orbits. We also describe $C^1$ -open sets of IFS’s admitting ergodic measures of positive entropy whose Lyapunov exponents along $M$ are all zero. The proofs involve the construction of non-hyperbolic measures for the induced IFS’s on the flag manifold.     

A sub-Riemannian curvature-dimension inequality,volume doubling property and the Poincaré inequality

Let $\mathbb M $ be a smooth connected manifold endowed with a smooth measure $\mu $ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$ , and which is symmetric with respect to $\mu $ . We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590, then the following properties hold:

• The volume doubling property;
• The Poincaré inequality;
• The parabolic Harnack inequality.

The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster–Tanaka–Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.     

A numerical study on exceptional eigenvalues of certain congruence subgroups of \mathrm {SO}(n,1) and \mathrm {SU}(n,1)

In a previous work, we applied lattice point theorems on hyperbolic spaces to obtain asymptotic formulas for the number of integral representations of negative integers by quadratic and Hermitian forms of signature \((n,1)\) lying in Euclidean balls of increasing radius. That formula involved an error term that depended on the first nonzero eigenvalue of the Laplace–Beltrami operator on the corresponding congruence hyperbolic manifolds. The aim of this paper is to compare the error term obtained by experimental computations with the error term mentioned above, for several choices of quadratic and Hermitian forms. Our numerical results provide evidence of the existence of exceptional eigenvalues for some arithmetic subgroups of \(\mathrm {SU}(3,1)\) , \(\mathrm {SU}(4,1)\) , and \(\mathrm {SU}(5,1)\) , and thus they contradict the generalized Selberg (and Ramanujan) conjecture in these cases. Furthermore, for several arithmetic subgroups of \(\mathrm {SO}(4,1)\) , \(\mathrm {SO}(6,1)\) , \(\mathrm {SO}(8,1)\) , and \(\mathrm {SU}(2,1)\) , there is evidence of a lower bound on the first nonzero eigenvalue that is better than the already known lower bound for congruences subgroups.     

Generalized quaternionic manifolds

We initiate the study of the generalized quaternionic manifolds by classifying the generalized quaternionic vector spaces, and by giving two classes of nonclassical examples of such manifolds. Thus, we show that any complex symplectic manifold is endowed with a natural (nonclassical) generalized quaternionic structure, and the same applies to the heaven space of any three-dimensional Einstein–Weyl space. In particular, on the product \(Z\) of any complex symplectic manifold \(M\) and the sphere, there exists a natural generalized complex structure, with respect to which \(Z\) is the twistor space of  \(M\) .     

{l_p}-Recovery of the Most Significant Subspace Among Multiple Subspaces with Outliers

We assume data sampled from a mixture of \(d\) -dimensional linear subspaces with spherically symmetric distributions within each subspace and an additional outlier component with spherically symmetric distribution within the ambient space (for simplicity, we may assume that all distributions are uniform on their corresponding unit spheres). We also assume mixture weights for the different components. We say that one of the underlying subspaces of the model is most significant if its mixture weight is higher than the sum of the mixture weights of all other subspaces. We study the recovery of the most significant subspace by minimizing the \(l_p\) -averaged distances of data points from \(d\) -dimensional subspaces of \(\mathbb R^D\) , where \(0 < p \in \mathbb R\) . Unlike other \(l_p\) minimization problems, this minimization is nonconvex for all \(p>0\) and thus requires different methods for its analysis. We show that if \(0 , then for any fraction of outliers, the most significant subspace can be recovered by \(l_p\) minimization with overwhelming probability (which depends on the generating distribution and its parameters). We show that when adding small noise around the underlying subspaces, the most significant subspace can be nearly recovered by \(l_p\) minimization for any \(0 with an error proportional to the noise level. On the other hand, if \(p>1\) and there is more than one underlying subspace, then with overwhelming probability the most significant subspace cannot be recovered or nearly recovered. This last result does not require spherically symmetric outliers.     

Metrics with prescribed Ricci curvature near the boundary of a manifold

Suppose $M$ is a manifold with boundary. Choose a point $o\in \partial M$ . We investigate the prescribed Ricci curvature equation $\mathop {\mathrm{Ric}}\nolimits (G)=T$ in a neighborhood of $o$ under natural boundary conditions. The unknown $G$ here is a Riemannian metric. The letter $T$ on the right-hand side denotes a (0,2)-tensor. Our main theorems address the questions of the existence and the uniqueness of solutions. We explain, among other things, how these theorems may be used to study rotationally symmetric metrics near the boundary of a solid torus $\mathcal{T }$ . The paper concludes with a brief discussion of the Einstein equation on $\mathcal{T }$ .