Curvature estimates for convex solutions of some fully nonlinear Hessian-type equations

Abstract

The curvature estimates of quotient curvature equation do not always exist even for convex setting [24]. Thus it is natural question to find other type of elliptic equations possessing curvature estimates. In this paper, we discuss the existence of curvature estimate for fully nonlinear elliptic equations defined by symmetric polynomials, mainlly, the linear combination of elementary symmetric polynomials.

Second order estimates for a class of complex Hessian equations on Hermitian manifolds

“…Such type of equations also originate in the study of J-equation on toric varieties by Collins-Székelyhidi [7]. The real analogous equation of (1.2) was studied by Li-Ren-Wang [23], and see Guan-Zhang [18] for interesting related work.…”

“…In this paper, we concentrate on the second order estimate for equation (1.2) with the condition g ∈ Γ k+1 (M ). We need to assume that Q satisfies the so called quotient concavity property introduced by Li-Ren-Wang [23].…”

“…Remark 1.3. For a non-negative constant α, Q = ασ k−1 (λ) + σ k (λ) satisfies the quotient concavity, see [23]. In general, Q may not be quotient concave.…”

“…In general, Q may not be quotient concave. A sufficient condition for Q to be quotient concave is given by Theorem 3 in [23].…”

A class of the non-degenerate complex quotient equations on compact Kähler manifolds

“…Recently, Guan-Zhang in [7] considered the (k − 1)-admissible solution without the sign of α k−1 and obtained the global C 2 estimates. Moreover, similar equations are also studied in [12]. Equations (1.1) include complex Monge-Ampère equation, the complex k−Hessian equations and (k, l)−quotient equations as special cases.…”