Abstract:
We prove uniform $L^p$ estimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding resul of [3] in the case of Laplace-- Beltrami operators on Riemannian manifolds. In doing so, we follow the methods, developed in [1] very closely. We also show that spectral regions in our $L^p$ resolvent estimates are optimal.

Abstract:
In this note we consider the Schr\"odinger equation on compact manifolds equipped with possibly degenerate metrics. We prove Strichartz estimates with a loss of derivatives. The rate of loss of derivatives depends on the degeneracy of metrics. For the non-degenerate case we obtain, as an application of the main result, the same Strichartz estimates as that in the elliptic case. This extends Strichartz estimates for Riemannian metrics proved by Burq-G\'erard-Tzvetkov to the non-elliptic case and improves the result by Salort. We also investigate the optimality of the result for the case on $\mathbb{S}^3\times \mathbb{S}^3$.

Abstract:
This paper is a self-contained presentation of certain aspects of the theory of weighted Sobolev spaces and elliptic operators on non-compact Riemannian manifolds. Specifically, we discuss (i) the standard and weighted Sobolev Embedding Theorems for general manifolds and (ii) Fredholm results for elliptic operators on manifolds with a finite number of ends modelled either on cones ("conifolds") or on cylinders. As an application of these results we present a detailed analysis of certain spaces of harmonic functions on conifolds. Some of the results presented here are of course well-known. Some others are probably known or self-evident to the experts. However, the current literature is not always easy to understand and is often sketchy, apparently not covering some aspects and consequences of the general theory which are useful in applications. In particular, in recent years results of this type have played an increasing role in Differential Geometry. The goal of this paper is thus to fill certain gaps in the current literature and to make these results available to a wider audience. The paper is also meant as a companion paper to the author's forthcoming articles [8],[9]. From this point of view it is still incomplete: future versions of this paper will incorporate more material, giving particular attention to uniform elliptic estimates for certain parametric connect sum constructions.

Abstract:
We determine lower and exact estimates of Kolmogorov, Gelfand and linear $n$-widths of unit balls in Sobolev norms in $L_{p}$-spaces on compact Riemannian manifolds. As it was shown by us previously these lower estimates are exact asymptotically in the case of compact homogeneous manifolds. The proofs rely on two-sides estimates for the near-diagonal localization of kernels of functions of elliptic operators.

Abstract:
This note is devoted to optimal spectral estimates for Schr\"odinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.

Abstract:
We consider the Dirac operator on compact quaternionic Kaehler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space.

Abstract:
The objective of the paper is to describe Besov spaces on general compact Riemannian manifolds in terms of the best approximation by eigenfunctions of elliptic differential operators.

Abstract:
In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds.

Abstract:
We consider quasicomplexes of pseudodifferential operators on a smooth compact manifold without boundary. To each quasicomplex we associate a complex of symbols. The quasicomplex is elliptic if this symbol complex is exact away from the zero section. We prove that elliptic quasicomplexes are Fredholm. Moreover, we introduce the Euler characteristic for elliptic quasicomplexes and prove a generalization of the Atiyah-Singer index theorem.

Abstract:
Along the lines of the classic Hodge-De Rham theory a general decomposition theorem for sections of a Dirac bundle over a compact Riemannian manifold is proved by extending concepts as exterior derivative and coderivative as well as as elliptic absolute and relative boundary conditions for both Dirac and Dirac Laplacian operators. Dirac sections are shown to be a direct sum of harmonic, exact and coexact spinors satisfying alternatively absolute and relative boundary conditions. Cheeger's estimation technique for spectral lower bounds of the Laplacian on differential forms is generalized to the Dirac Laplacian. A general method allowing to estimate Dirac spectral lower bounds for the Dirac spectrum of a compact Riemannian manifold in terms of the Dirac eigenvalues for a cover of 0-codimensional submanifolds is developed. Two applications are provided for the Atiyah-Singer operator. First, we prove the existence on compact connected spin manifolds of Riemannian metrics of unit volume with arbitrarily large first non zero eigenvalue. Second, we prove that on a degenerating sequence of oriented, hyperbolic, three spin manifolds for any choice of the spin structures the first positive non zero eigenvalue is bounded from below by a positive uniform constant.