A class of prescribed Weingarten curvature equations in Euclidean space

Prescribed Weingarten curvature equations in warped product manifolds

“…(2) Clearly, our Corollary 1.3 covers Chen-Shang-Tu's and Shang-Tu's main results in [2,27] (mentioned in (3) of Remark 1.1) as special cases.…”

“…where u i = g ij u j = δ ij u j and |Du| 2 = u i u i . Of course, in this paper we use the Einstein summation convention -repeated superscripts and subscripts should be made summation 2 . The outward unit normal vector field of G is given by…”

“…for some unknown function u : S n → R defined over S n , where α l (x), 0 ≤ l ≤ k − 2, are positive functions; Fu-Yau [6,7] proposed an equation of this type in the study of the Hull-Strominger system in theoretical physics; Phong-Picard-Zhang investigated the Fu-Yau equation and its generalization in series works [24,25,26]. Recently, inspired by Krylov's and Guan-Zhang's works [11,14], Chen-Shang-Tu [2] considered the following equation…”

The Dirichlet Problem for mixed Hessian type equations on Riemannian manifolds

“…Later the Eq. (1.3) in Neumann problem, prescribed curvature problem, complex manifolds were also discussed in [4,5,33,7,6,31,32,22].…”

The Dirichlet problem for mixed Hessian equations on Hermitian manifolds

“…Then they obtained a priori C 2 estimate of the (k − 1)-admissible solution of equation (1.4) without sign requirement for α k−1 and solved the Dirichlet problem for the corresponding equation. Later the corresponding Neumann problem and prescribed curvature equations were also discussed in [4,5,45,7]. Recently, Zhang [43] cosidered the Dirichlet problem for (1.3) on complex domains in C n , which can be seen as the extension of Guan-Zhang's result.…”