A note on star-shaped compact hypersurfaces with prescribed scalar curvature in space forms

Abstract

Abstract. In [7], Guan, Ren and Wang obtained a C 2 a priori estimate for admissible 2-convex hypersurfaces satisfying the Weingarten curvature equation σ 2 (κ(X)) = f (X, ν(X)). In this note, we give a simpler proof of this result, and extend it to space forms.

The Fu–Yau equation with negative slope parameter

“…We present the argument for the sake of completeness. Since we are dealing with a 2-Hessian equation, we will also use ideas from [30]. Before proceeding in cases, we use the critical equation DG = 0 to notice the following estimate which holds for each fixed index i,…”

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

“…In [11], Chen-Li-Wang extended these estimates to non Codazzi case in warped product space. In [44], Spruck-Xiao extended 2-convex case in [24] to space forms and give a simple proof for the Euclidean case. We also note the recently important work on the curvature estimates and C 2 estimates developed by Guan [16] and Guan-Spruck-Xiao [26].…”

Curvature estimates for immersed hypersurfaces in Riemannian manifolds

“…The curvature estimate was established recently by Guan-Ren-Wang [15] for convex hypersurfaces when F in (1.6) is an elementary symmetric function σ k , 1 ≤ k ≤ n. It's also proved in [15] that, curvature estimate holds for general starshaped admissible solutions of equation (1.6) when F = σ 2 . Spruck-Xiao [34] subsequently found a very nice simplified proof of the estimate in [15] for F = σ 2 , their estimate is also valid in general space form. On the other hand, estimates obtained in [4,8,15,34] depend also on the lower bound of f .…”

“…Spruck-Xiao [34] subsequently found a very nice simplified proof of the estimate in [15] for F = σ 2 , their estimate is also valid in general space form. On the other hand, estimates obtained in [4,8,15,34] depend also on the lower bound of f . Estimate (1.5) does not depend on the lower bound of σ 2 (κ).…”