Interior C2 regularity of convex solutions to prescribing scalar curvature equations

Abstract

We establish interior C 2 estimates for convex solutions of scalar curvature equation and σ2-Hessian equation. We also prove interior curvature estimate for isometrically immersed hypersurfaces (M n , g) ⊂ R n+1 with positive scalar curvature. These estimates are consequences of an interior estimates for these equations obtained under a weakened condition.

Interior Hessian estimates for Sigma-2 equations in dimension three

“…The purely interior C 2 estimates for solutions of equations (1.1) and (1.2) with certain convexity constraints were obtained recently by McGonagle-Song-Yuan in [14] and Guan and the author in [5]. Now we state our main result in this paper Theorem 1.…”

Strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms

“…A central issue is to provide certain uniform C 2 bound for u ǫ . Motivated by a recent work of Guan-Qiu [18], we obtain interior C 2 estimates for strictly locally convex solutions to prescribed scalar curvature equations in H n+1 , which, together with Evans-Krylov interior estimates (see [6,20]) and standard diagonal process, lead to the following existence result. Theorem 1.10.…”

“…The convexity of solutions is a very important prerequisite in this paper, due to the following two reasons: first, the C 2 boundary estimates derived in section 3 require the condition of convexity; second, the C 2 interior estimates for prescribed scalar curvature equations in section 6 need certain convexity assumption (see [18]). Therefore, the preservation of convexity of solutions is vital in order to perform the continuity process.…”

“…As in [18], we divide our discussion into three cases. We show all the details to indicate the tiny differences due to the outer space H n+1 .…”

Smooth Solutions to Asymptotic Plateau Type Problem in Hyperbolic Space

“…An immediate trial is to establish pure interior curvature estimate, which is only possible when k ≤ 2 (see the counterexamples of Pogorelov [10] and Urbas [15] when k ≥ 3). In [12], we applied the idea of Guan-Qiu [9] to derive interior curvature estimate for strictly locally convex solutions to prescribed scalar curvature equation in hyperbolic space, and consequently, we proved the existence of a smooth hypersurface to (1.1) when k = n = 2. The next trial is to establish Pogorelov type interior curvature estimate.…”