We show that if M is a complete Riemannian manifold and H = ∆ + V is a Schrödinger operator, then the existence of a positive solution of Hu = 0 outside a compact set is equivalent to the finiteness of the Morse index of H .

We prove that a riemannian metric on the 2-sphere or the projective plane can be C 2 approximated by a C ∞ metric whose geodesic flow has an elliptic closed geodesic. In this paper we show how to overcome a difficulty presented by Henri Poincaré in the C 2 generic case. In 1905, H. Poincaré [34, p. 259] claimed that any convex surface in R 3 should have an elliptic or degenerate non self-inters...

We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r = 2 and r = 3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds for r = 2, 3 and 4 respectively...

This is a survey on the convergence theory developed rst by Cheeger and Gromov. In their theory one is concerned with the compactness of the class of riemannian manifolds with bounded curvature and lower bound on the injectivity radius. We explain and give proofs of almost all the major results, including Anderson's generalizations to the case where all one has is bounded Ricci curvature. The e...

In this paper, we establish the uniqueness of heat flow of harmonic maps into (N,h) that have sufficiently small renormalized energies, provided that N is either a unit sphere Sk−1 or a Riemannian homogeneous manifold. For such a class of solutions, we also establish the convexity property of the Dirichlet energy for t ≥ t0 > 0 and the unique limit property at time infinity. As a corollary, the...

Suppose M is a closed Riemannian manifold. For $$C^2$$ generic (in the sense of Mañé) Tonelli Hamiltonian $$H: T^*M\rightarrow \mathbb {R}$$ , minimal viscosity solution $$u_\lambda ^-:M\rightarrow negative discounted equation $$\begin{aligned} -\lambda u+H(x,d_xu)=c(H),\quad x\in M,\ \lambda >0 \end{aligned}$$ with Mañé’s critical value c(H) converges to uniquely established $$u_0^-$$ Hamilton...

The Lichnerowicz conjecture asserted that every harmonic Riemannian manifold is locally isometric to a two-point homogeneous space. In 1992, E. Damek and F. Ricci produced a family of counter-examples to this conjecture, which arise as abelian extensions of two-step nilpotent groups of type-H. In this paper we consider a broader class of Riemannian manifolds: solvmanifolds of Iwasawa type with ...

Let G=H be a semisimple globally hyperbolic symmetric space and let ' be a H-spherical function on G=H. We derive an expansion formula for ' similar to the Harish-Chandra formula for spherical functions on a Riemannian symmetric space. We use this result to analytically continuate the spherical functions in the parameters. A functional equation for ' is derived and then used to invert the spher...