Abstract. In this paper, we use the inverse curvature flow to prove a sharp geometric inequality on star-shaped and two-convex hypersurface in hyperbolic space.

We prove that in Riemannian manifolds the k-th Steklov eigenvalue on a domain and the square root of the k-th Laplacian eigenvalue on its boundary can be mutually controlled in terms of the maximum principal curvature of the boundary under sectional curvature conditions. As an application, we derive a Weyl-type upper bound for Steklov eigenvalues. A Pohozaev-type identity for harmonic functions on the domain and the min-max variational characterization of both eigenvalues are important ingredients.

In this paper, firstly, inspired by Natário's recent work [27], we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space H n+1 and in the sphere S n+1 . We also get the rigidity in the spherical case. Secondly, we use the inverse mean curvature flow in sphere [12,26] to prove an optimal Sobolev type inequality for closed convex hypersurfaces in the sphere.

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