Interior \(C^2\) regularity of convex solutions to prescribing scalar curvature equations. (English) Zbl 1429.35086
Suppose that \(\mathcal{M}=(M^n,g)\) is an isometrically immersed hypersurface in \(\mathbb{R}^{n+1}\) which is a graph over a ball of radius \(r\) in \(\mathbb{R}^n\). Under the assumption that the scalar curvature \(R_g\) of \(\mathcal{M}\) is positive, the author estimates the principal curvatures \(\kappa_i\) of \(\mathcal{M}\) on the ball of radius \(\frac12r\) in the form \[ \sup_{B_{\frac12r}}|\kappa_i|\le C\big(\|g\|_{C^4(B_r)},\inf_{B_r}R_g,\|M\|_{C^1(B_r)}\big)\,. \] This provides local interior curvature estimates in this context and generalizes E. Heinz’s [J. Anal. Math. 7, 1–52 (1959; Zbl 0152.30901)] interior estimate and is, in particular, independent of boundary data. This is related to finding interior \(C^2\) estimates for solutions to tprescribing scalar curvature equation and \(\sigma_2\)-Hessian equation. The authors obtain appropriate results in this context as well.
Reviewer: Peter B. Gilkey (Eugene)
MSC:
| 35J60 | Nonlinear elliptic equations |
| 58J05 | Elliptic equations on manifolds, general theory |
| 35B45 | A priori estimates in context of PDEs |
Keywords:
scalar curvature equation; \(\sigma_2\)-Hessian equation; interior estimates; a priori regularity estimatesCitations:
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