Sharp bounds for the first eigenvalue of a fourth order Steklov problem

We study the biharmonic Steklov eigenvalue problem on a compact Riemannian manifold Ω with smooth boundary. We give a computable, sharp lower bound of the first eigenvalue of this problem, which depends only on the dimension, a lower bound of the Ricci curvature of the domain, a lower bound of the mean curvature of its boundary and the inner radius. The proof is obtained by estimating the isoperimetric ratio of non-negative subharmonic functions on Ω, which is of independent interest. We also give a comparison theorem for geodesic balls

Branson's Q-curvature in Riemannian and Spin Geometry

On a closed 4-dimensional Riemannian manifold, we give a lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson's Q-curvature. As a consequence, if the manifold is spin, we relate the first eigenvalue of the Dirac operator to the total Branson's Q-curvature. On a closed n-dimensional manifold, $n\ge 5$, we compare the three basic conformally covariant operators : the Branson-Paneitz, the Yamabe and the Dirac operator (if the manifold is spin) through their first eigenvalues. Equality cases are also characterized.Comment: 14 pages, Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branso

Rigidit\'e conforme des h\'emisph\`eres S^4_+ et S^6_+

Let (M,g) be a four or six dimensional compact Riemannian manifold which is locally conformally flat and assume that its boundary is totally umbilical. In this note, we prove that if the Euler characteristic of M is equal to 1 and if its Yamabe invariant is positive, then (M,g) is conformally isometric to the standard hemisphere. As an application and using a result of Hang-Wang, we prove a rigidity result for these hemispheres regarding the Min-Oo conjecture.Comment: 8 pages, to appear in Mathematische Zeitschrif

Rigidity of Compact Riemannian Spin Manifolds with Boundary

In this article, we prove new rigidity results for compact Riemannian spin manifolds with boundary whose scalar curvature is bounded from below by a nonpositive constant. In particular, we obtain generalizations of a result of Hang-Wang (Pac J Math 232(2):283-288, 2007) based on a conjecture of Schroeder and Strake (Comment Math Helv 64:173-186, 1989

On a spin conformal invariant on manifolds with boundary

Let M be an n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric g, a spin structure σ and a chirality operator Γ. We define and study some properties of a spin conformal invariant given by: $$\lambda_{\rm min}(M, \partial M) := \underset{\overline{g}\in[g]}{\rm inf}|\lambda_1^\pm(\overline{g})|{\rm Vol}(M, \overline{g})^{\frac{1}{n}},$$ where $$\lambda_1^\pm(\overline{g})$$ is the smallest eigenvalue of the Dirac operator under the chiral bag boundary condition $${\mathbb{B}}^\pm_{\overline{g}}$$ . More precisely, we show that if n ≥2 then: $$\lambda_{\rm min}(M, \partial M) \leq \lambda_{\rm min}({\mathbb{S}}_+^n, \partial{\mathbb{S}}_+^n).$

Nonexistence of DEC spin fill-ins

In this note, we show that a closed spin Riemannian manifold does not admit a spin fill-in satisfying the dominant energy condition (DEC) if a certain generalized mean curvature function is point-wise large

A spinorial proof of the rigidity of the Riemannian Schwarzschild manifold

We revisit and generalize a recent result of Cederbaum [C2, C3] concerning the rigidity of the Schwarzschild manifold for spin manifolds. This includes the classical black hole uniqueness theorems [BM, GIS, Hw] as well as the more recent uniqueness theorems for pho-ton spheres [C1, CG1, CG2]