The total Cartan curvature of Reinhardt surfaces

The Cartan umbilical tensor is a crucial biholomorphic invariant for threedimensional strictly pseudoconvex CR manifolds. When considering compact manifolds, its L1-norm, referred to as the total Cartan curvature, is an important global numerical invariant. In this study, we seek to calculate the total Cartan curvature of T3, the three-torus, which features a specific T2-invariant CR structure, incorporating Reinhardt boundaries with closed logarithmic generating curves as a special instance. Our findings present an unexpected corollary indicating that, for many of these boundaries, the total Cartan curvature is equal to 8 & pi;2 times the Burns-Epstein invariant.& COPY; 2023 Elsevier Inc. All rights reserved.