This paper is devoted to the study of the compact hyperbolic 3-manifolds uniformized by the Fibonacci groups. It is shown that their volumes are equal to volumes of non-compact hyperbolic 3-manifolds arising as complements of some well-known knots. All these volumes are described in terms of the Lobachevsky function.

Abstract. We first give an elementary proof of a result relating the eigenvalues of the Dirac operator to Branson’s Q-curvature on 4-dimensional spin compact manifolds. In the case of n-dimensional closed compact (spin) manifolds we then use the conformal covariance of the Dirac, Yamabe and Branson-Paneitz operators to compare appropriate powers of their first eigenvalues. Equality cases are al...

In this article we collect a series of observations that constrain actions of many groups on compact manifolds. In particular, we show that “generic” finitely generated groups have no smooth volume preserving actions on compact manifolds while also producing many finitely presented, torsion free groups with the same property.

In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type dr , r > 1, where d is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction.

We study the geometry of compact Lorentzian manifolds that admit a somewhere timelike Killing vector field, and whose isometry group has infinitely many connected components. Up to a finite cover, such manifolds are products (or amalgamated products) of a flat Lorentzian torus and a compact Riemannian (resp., lightlike) manifold.

We prove the existence of at least two timelike non self-intersecting periodic geodesics in compact stationary Lorentzian manifolds and we discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a stationary Lorentzian metric if and only if M admits a smooth circle action without fixed points.

Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete conformally compact metrics on such manifolds. In addition, it is shown that the Ricci curvature forms an elliptic system in geodesic-harmonic coordinates naturally associated with the boundary data.

In this paper, we prove that the Hodge-Laplace operator on strongly pseudoconvex compact complex Finsler manifolds is a self-adjoint elliptic operator. Then, from the decomposition theorem for self-adjoint elliptic operators, we obtain a Hodge decomposition theorem on strongly pseudoconvex compact complex Finsler manifolds. M.S.C. 2010: 53C56, 32Q99.

The Hodge theorem and Hirzebruch signature theorem form an important bridge between geometric and topological properties of compact smooth manifolds. There has been a great deal of work over the past thirty years aimed at understanding how to generalize these theorems to L2 results in the noncompact and singular settings. Early and important work was done by Atiyah, Patodi and Singer [2]. Their...

In [26], Perelman established a differential Li-YauHamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds (also see [23]). As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these res...