On the Chern–Moser–Weyl tensor of real hypersurfaces

Abstract

We derive an explicit formula for the well-known Chern-Moser-Weyl tensor for nondegenerate real hypersurfaces in complex space in terms of their defining functions. The formula is considerably simplified when applying to "pluriharmonic perturbations" of the sphere or to a Fefferman approximate solution to the complex Monge-Ampère equation. As an application, we show that the CR invariant one-form Xα constructed recently by Case and Gover is nontrivial on each real ellipsoid of revolution in C 3 , unless it is equivalent to the sphere. This resolves affirmatively a question posed by these two authors in 2017 regarding the (non-) local CR invariance of the I ′ -pseudohermitian invariant in dimension five and provides a counterexample to a recent conjecture by Hirachi.

The holomorphic sectional curvature and “convex” real hypersurfaces in Kähler manifolds

“…The Gauss equation (2.9) is similar to several versions in the literature (cf. [5]) and has found several applications [14,12]. For example, Reiter and the author [12] used this equation to establish an explicit and concise formula for the well-known Chern-Moser-Weyl tensor.…”

“…[5]) and has found several applications [14,12]. For example, Reiter and the author [12] used this equation to establish an explicit and concise formula for the well-known Chern-Moser-Weyl tensor. On the other hand, equation (2.10), which has no Riemannian counterpart, relates the (intrinsic) torsion of the submanifold and the (extrinsic) second fundamental form of the immersion.…”

“…Proof. The last assertion follows from explicit formulas for the second fundamental form in [14] and [12]. Notice that the Levi matrix h α β in the frame Z α := w ∂ α − α ∂ w is given by…”

“…where Z Z is the Hermitian part of the Hessian of . Following [12], we define the second order differential operator…”

“…To show that inequalities (1.8) and (1.7) are sharp, we consider the following real hypersurface which has been studied in various papers, e.g., [12] and the references therein. Precisely, let…”