A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds

Evolution of the first eigenvalue of buckling problem on Riemannian manifold under Ricci flow

Among the eigenvalue problems of the Laplacian, the biharmonic operator eigenvalue problems are interesting projects because these problems root in physics and geometric analysis. The buckling problem is one of the most important problems in physics, and many studies have been done by the researchers about the solution and the estimate of its eigenvalue. In this paper, first, we obtain the evol...

Kähler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz

A universal lower bound for the first eigenvalue of the Dirac operator on quaternionic Kähler manifolds

A universal lower bound for the first positive eigenvalue of the Dirac operator on a compact quaternionic Kähler manifold M of positive scalar curvature is calculated. It is shown that it is equal to the first positive eigenvalue on the quaternionic projective space. For this, the horizontal tangent bundle on the canonical SO(3)-bundle over M is equipped with a hyperkählerian structure and the ...

Eigenvalue Comparison on Bakry-emery Manifolds

It is called shrinking, steady, or expanding soliton if a > 0, a = 0 or a < 0 respectively. More generally (M, g, f) is called a Bakry-Emery manifold if the so-called Bakry-Emery Ricci tensor Rcij + fij ≥ agij for some a ∈ R. In this paper we apply the modulus of continuity estimates developed in [AC,AC2,AC3] to give a different proof of an eigenvalue comparison estimate on Bakry-Emery manifold...

Compactness of the Space of Embedded Minimal Surfaces with Free Boundary in Three-manifolds with Nonnegative Ricci Curvature and Convex Boundary

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact n-dimensional Riemannian manifold which has nonnegative Ricci curvature and strictly convex boundary. When n “ 3, this implies an apriori curvature estimate for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. A...