Spectral rigidity of compact Kaehler and contact manifolds

Spectral properties of Schrödinger operators on compact manifolds: rigidity, flows, interpolation and spectral estimates

This note is devoted to optimal spectral estimates for Schrödinger 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.

Lecture on Kaehler manifolds

Rigidity and Other Topological Aspects of Compact Nonpositively Curved Manifolds

Let M be a compact connected Riemannian manifold whose sectional curvature values are all nonpositive. Let T denote the fundamental group of M. We prove that any homotopy equivalence ƒ : N —> M from a compact closed manifold TV is homotopic to a homeomorphism, provided that m > 5 where m = dim M . We show that the surgery L-group Lk+m(T, w{) is isomorphic to the set of homotopy classes of k k m...

Rigidity of Compact Manifolds with Boundary and Nonnegative Ricci Curvature

Let Ω be an (n + 1)-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary M = ∂Ω. Assume that the principal curvatures of M are bounded from below by a positive constant c. In this paper, we prove that the first nonzero eigenvalue λ1 of the Laplacian of M acting on functions on M satisfies λ1 ≥ nc2 with equality holding if and only if Ω is isometric to a...

Nonlinear Flows and Rigidity Results on Compact Manifolds

Abstract. This paper is devoted to rigidity results for some elliptic PDEs and related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution than the constant one at least when a parameter is in a certain range. This parameter can be used as an estimate for the best constant in the corres...