We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of Laplace-Beltrami operator in some Riemannian manifold. These domains are close to geodesic spheres of small radius centered at a nondegenerate critical point of the scalar curvature.
New examples of extremal Kähler metrics are given on blow-ups of parabolic ruled surfaces. The method used is based on the gluing construction of Arezzo, Pacard and Singer [5]. This enables to endow ruled surfaces of the form with special parabolic structures such that the associated iterated blow-up admits an extremal metric of non-constant scalar curvature.
We provide a new proof of a result of X.X. Chen and G.Tian [5]: for a polarized extremal Kähler manifold, the minimum of the modified K-energy is attained at an extremal metric. The proof uses an idea of C. Li [16] adapted to the extremal metrics using some weighted balanced metrics.
A complete solution to the quaternionic contact Yamabe problem on the seven-dimensional sphere is given. Extremals for the Sobolev inequality on the seven-dimensional Heisenberg group are explicitly described and the best constant in the L2 Folland–Stein embedding theorem is determined.
We give the definition of -biminimal submanifolds and derive the equation for -biminimal submanifolds. As an application, we give some examples of -biminimal manifolds. Finally, we consider -minimal hypersurfaces in the product space and derive two rigidity theorems.