Let (M, g) be a compact Riemannian manifold of dimension ≥ 3. We show that there is a metrics g̃ conformal to g and of volume 1 such that the first positive eigenvalue the conformal Laplacian with respect to g̃ is arbitrarily large. A similar statement is proven for the first positive eigenvalue of the Dirac operator on a spin manifold of dimension ≥ 2.

We estimate the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature in terms of the diameter and the lower bound of Ricci curvature and give an affirmative answer to the conjecture of H. C. Yang for the first Dirichlet eigenvalue.

We consider the minimization problem min Ω∈X (Λ2 − Λ∞) (Ω), where Λ2(Ω) and Λ∞(Ω) are the (square root of the) first eigenvalue of the Laplacian and the first eigenvalue of the ∞−Laplacian respectively. X is the class of convex domains with prescribed diameter. We prove existence of a solution, and we provide several geometrical properties of minimizers.

We consider a shape optimization problem for the first mixed Steklov–Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. give geometric proof which is motivated Newton’s shell theorem.