Let (M,ω _g) be a complete Kähler manifold of complex dimension n. We prove that if the holomorphic sectional curvature satisfies HSC 2 , then the first positive eigenvalue λ _1 of the Laplacian on (M,ω _g) satisfies λ _1 320(n-1)+576/81(n-1)+144. This result is established through a new Bochner–Kodaira type identity specifically developed for holomorphic sectional curvature.
In this paper, we establish the existence of a strictly convex hypersurface function u defined in Ω that satisfies k-Hessian curvature equations subject to the second boundary condition Du(Ω )=Ω ^* , where Ω and Ω ^* are smooth strictly convex bounded domains in ℝ^n . By utilizing the orthogonal invariance of hypersurface, we employ special vector fields generated by the infinitesimal rotations in ℝ^n+1 to establish the boundary C^2 estimates. Our work extends the results of Urbas on the second boundary value problem of Weingarten curvature equations [31] and Hessian equations [30].
In this paper we consider the obstacle problem for the elastic flow and deduce an adapted version of the Energy Dissipation Inequality. The inequality benefits from the improved regularity of weak solutions, which is another main result of this paper. We also derive the uniqueness of weak solutions as an application of the improved regularity.
