Interior \(C^2\) regularity of convex solutions to prescribing scalar curvature equations. (English) Zbl 1429.35086

Guan, Pengfei; Qiu, Guohuan

Duke Math. J. 168, No. 9, 1641-1663 (2019).

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 estimates

Citations:

Zbl 0152.30901

References:

[1]	L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), no. 3-4, 261-301. · Zbl 0654.35031
[2]	C. Chen, Optimal concavity of some hessian operators and the prescribed \(\sigma_2\) curvature measure problem, Sci. China Math. 56 (2013), no. 3, 639-651. · Zbl 1276.35081
[3]	C. Chen, F. Han, and Q. Ou, The interior \(C^2\) estimate for the Monge-Ampère equation in dimension \(n=2\), Anal. PDE 9 (2016), no. 6, 1419-1432. · Zbl 1353.35175
[4]	P. Guan, P. Lu, and Y. Xu, A rigidity theorem for codimension one shrinking gradient Ricci solitons in \(\mathbb{R}^{n+1} \), Calc. Var. Partial Differential Equations 54 (2015), no. 4, 4019-4036. · Zbl 1335.53060
[5]	P. Guan and S. Lu, Curvature estimates for immersed hypersurfaces in Riemannian manifolds, Invent. Math. 208 (2017), no. 1, 191-215. · Zbl 1361.53017
[6]	P. Guan, C. Ren, and Z. Wang, Global \(C^2\)-estimates for convex solutions of curvature equations, Comm. Pure Appl. Math. 68 (2015), no. 8, 1287-1325. · Zbl 1327.53009
[7]	E. Heinz, On elliptic Monge-Ampère equations and Weyl’s embedding problem, J. Analyse Math. 7 (1959), 1-52. · Zbl 0152.30901
[8]	M. Lin and N. S. Trudinger, On some inequalities for elementary symmetric functions, Bull. Austral. Math. Soc. 50 (1994), no. 2, 317-326. · Zbl 0855.26006 · doi:10.1017/S0004972700013770
[9]	M. Mcgonagle, C. Song, and Y. Yuan, Hessian estimates for convex solutions to quadratic Hessian equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 2, 451-454. · Zbl 1423.35107
[10]	L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337-394. · Zbl 0051.12402
[11]	A. V. Pogorelov, Extrinsic Geometry of Convex Surfaces, Transl Math. Monogr. 35, Amer. Math. Soc. Providence, 1973. · Zbl 0311.53067
[12]	A. V. Pogorelov, The Minkowski Multidimensional Problem, V. H. Winston, Washington, DC, 1978. · Zbl 0387.53023
[13]	G. Qiu, Interior Hessian estimates for sigma-2 equations in dimension three, preprint, 2017.
[14]	J. Urbas, On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations, Indiana Univ. Math. J. 39 (1990), no. 2, 355-382. · Zbl 0724.35028
[15]	M. Warren and Y. Yuan, Hessian estimates for the \(\sigma_2\)-equation in dimension \(3\), Comm. Pure Appl. Math. 62 (2009), no. 3, 305-321. · Zbl 1173.35388