On the topology of moduli spaces of non-negatively curved Riemannian metrics

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist infinite sequences of closed (respectively, open) manifolds of pairwise distinct homotopy type for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics.

The Injectivity Theorem on a Non-Compact Kähler Manifold

In this paper, we establish an injectivity theorem on a weakly pseudoconvex Kähler manifold X with negative sectional curvature. For this purpose, we develop the harmonic theory in this circumstance. The negative sectional curvature condition is usually satisfied by the manifolds with hyperbolicity, such as symmetric spaces, bounded symmetric domains in Cn, hyperconvex bounded domains, and so on.

Hardy–Adams Inequalities on ℍ2 × ℝ n-2

Let ℍ 2 {\mathbb{H}^{2}} be the hyperbolic space of dimension 2. Denote by M n = ℍ 2 × ℝ n - 2 {M^{n}=\mathbb{H}^{2}\times\mathbb{R}^{n-2}} the product manifold of ℍ 2 {\mathbb{H}^{2}} and ℝ n - 2 ( n ≥ 3 ) {\mathbb{R}^{n-2}(n\geq 3)} . In this paper we establish some sharp Hardy–Adams inequalities on M n {M^{n}} , though M n {M^{n}} is not with strictly negative sectional curvature. We also show that the sharp constant in the Poincaré–Sobolev inequality on M n {M^{n}} coincides with the best Sobolev constant, which is of independent interest.

Subconvex bounds for Hecke–Maass forms on compact arithmetic quotients of semisimple Lie groups

Let H be a semisimple algebraic group, K a maximal compact subgroup of $$G:=H({{\mathbb {R}}})$$ G : = H ( R ) , and $$\Gamma \subset H({{\mathbb {Q}}})$$ Γ ⊂ H ( Q ) a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke–Maass forms on the locally symmetric space $$\Gamma \backslash G/K$$ Γ \ G / K to corresponding bounds on the arithmetic quotient $$\Gamma \backslash G$$ Γ \ G for cocompact lattices using the spectral function of an elliptic operator. The bounds obtained extend known subconvex bounds for automorphic forms to non-trivial K-types, yielding such bounds for new classes of automorphic representations. They constitute subconvex bounds for eigenfunctions on compact manifolds with both positive and negative sectional curvature. We also obtain new subconvex bounds for holomorphic modular forms in the weight aspect.

Prime Geodesic Theorems for Compact Locally Symmetric Spaces of Real Rank One

Our basic objects will be compact, even-dimensional, locally symmetric Riemannian manifolds with strictly negative sectional curvature. The goal of the present paper is to investigate the prime geodesic theorems that are associated with this class of spaces. First, following classical Randol’s appraoch in the compact Riemann surface case, we improve the error term in the corresponding result. Second, we reduce the exponent in the newly acquired remainder by using the Gallagher–Koyama techniques. In particular, we improve DeGeorge’s bound Oxη, 2ρ − ρn ≤ η < 2ρ up to Ox2ρ−ρηlogx−1, and reduce the exponent 2ρ − ρn replacing it by 2ρ − ρ4n+14n2+1 outside a set of finite logarithmic measure. As usual, n denotes the dimension of the underlying locally symmetric space, and ρ is the half-sum of the positive roots. The obtained prime geodesic theorem coincides with the best known results proved for compact Riemann surfaces, hyperbolic three-manifolds, and real hyperbolic manifolds with cusps.

On Asymptotic Behavior of Zeta Singularities for Compact Locally Symmetric Spaces

We obtain precise estimates for the number of singularities of Selberg’s and Ruelle’s zeta functions for compact, higher-dimensional, locally symmetric Riemannian manifolds of strictly negative sectional curvature. The methods applied in this research represent a generalization of the methods described in the case of a compact Riemann surface. In particular, this includes an application of the Phragmen-Lindelof theorem, the variation of the argument of certain zeta functions, as well as the use of some classical analytic number theory techniques.

The Bourguignon Laplacian and Harmonic Symmetric Bilinear Forms

In this paper, we study the kernel and spectral properties of the Bourguignon Laplacian on a closed Riemannian manifold, which acts on the space of symmetric bilinear forms (considered as one-forms with values in the cotangent bundle of this manifold). We prove that the kernel of this Laplacian is an infinite-dimensional vector space of harmonic symmetric bilinear forms, in particular, such forms on a closed manifold with quasi-negative sectional curvature are zero. We apply these results to the description of surface geometry.