In this paper, we study three types of rotational surfaces in Galilean 3-spaces. We classify satisfying $$L_1G=F(G+C)$$ for some constant vector $C\in \mathbb{G}^3$ and smooth function $F$, where $L_1$ denotes the Cheng-Yau operator.

In this paper, by modifying Cheng-Yau$'$s technique to complete hypersurfaces in $S^{n+1}(1)$, we prove a rigidity theorem under the hypothesis of the mean curvature and the normalized scalar curvature being linearly related which improve the result of [H. Li, Hypersurfaces with constant scalar curvature in space forms, {em Math. Ann.} {305} (1996), 665--672].  

The quadrics are all surfaces that can be expressed as a second degree polynomialin x, y and z. We study the Gauss map G of quadric surfaces in the 3-dimensional Euclidean space R^3 with respect to the so called L_1 operator ( Cheng-Yau operator □) acting on the smooth functions defined on the surfaces. For any smooth functions f defined on the surfaces, L_f=tr(P_1o hessf), where P_1 is t...

We give an elementary proof to the asymptotic expansion formula of Rochon and Zhang (2012) for unique complete Kähler-Einstein metric Cheng Yau (1980), Kobayashi (1984), Tian (1987) Bando (1990) on quasi-projective manifolds. The main tools are solution second-order ordinary differential equations (ODEs) with constant coefficients spectral theory Laplacian operator a closed manifold.

We prove some inequalities of Payne-P\'olya-Weinberger-Yang type for eigenvalues fourth-order elliptic operators in weighted divergence form on complete Riemannian manifolds which generalizes the corresponding result clamped plate problem. also estimates lower order that contain from literature. As an application our results, we obtain bi-drifted Cheng-Yau operator.

in this paper, by modifying cheng-yau$'$s technique to complete hypersurfaces in $s^{n+1}(1)$, we prove a rigidity theorem under the hypothesis of the mean curvature and the normalized scalar curvature being linearly related which improve the result of [h. li, hypersurfaces with constant scalar curvature in space forms, {em math. ann.} {305} (1996), 665--672].

In this paper, we compute universal estimates of eigenvalues a coupled system elliptic differential equations in divergence form on bounded domain Euclidean space. As an application, show interesting case rigidity inequalities the Laplacian, more precisely, consider countable family domains Gaussian shrinking soliton that makes behavior known Laplacian invariant by first-order perturbation Lapl...