A constant rank theorem for solutions of fully nonlinear elliptic equations

Convexity is an important geometric property associated with the study of partial differential equations, in particular for equations related to problems in differential geometry. There is a vast literature on this subject. In an important development in 1985, a technique was devised to deal with the convexity issue via the homotopy method of deformation in the work of Caffarelli and Friedman [7]. In [7], the existence of convex solutions for semilinear elliptic equations in two dimensions was proved by a form of deformation lemma using the strong maximum principle (see also the work of Singer, Wong, Yau, and Yau [17] for a similar approach). The core of this approach is the establishment of the constant rank theorem; that is, the rank of the Hessian of the corresponding convex solution is constant. The result in [7] was later generalized to higher dimensions in [15]. The constant rank theorem is a refined statement of convexity. This has profound implications in the geometry of solutions. The idea of the deformation lemma and the establishment of the constant rank theorem can be extended to various nonlinear differential equations in differential geometry involving symmetric curvature tensors. Recently, in connection to the Christoffel-Minkowski problem and the problem of prescribing Weingarten curvatures in classical differential geometry, this form of deformation lemma was extended to some equations involving the second fundamental forms of embedded hypersurfaces in R [11, 12, 13]. The constant rank theorem shares similar geometric flavors in spirit with a classical theorem of Hartman and Nirenberg [14], where they treated hypersurfaces in R with a vanishing spherical Jacobian. A pertinent question is under what structural conditions for partial differential equations is the positivity of the symmetric curvature tensor preserved under homotopy deformation? The purpose of this paper is to establish a general principle