Lower bounds for the mean curvature of hollow tubes around complex hypersurfaces and totally real submanifolds
In [Compos. Math. 86, No. 3, 317-335 (1993; Zbl 0782.53054)]\ the authors derived comparison theorems for the mean curvature of tubular hypersurfaces in complete Kähler manifolds, taking as models the tubular hypersurfaces around a complex projective space embedded as a totally geodesic submanifold in a complex projective space \(\mathbb{C} P^n\). The paper under review continues this work. Here the authors derive comparison theorems for the mean curvature of a tubular hypersurface about a compact commplex hypersurface \(P\) and about a compact totally real submanifold \(P\) of dimension \(n\) in a complex Kähler manifold \(M^{2n}\), with some specific curvature bounds, when the models for the comparison are the tubular hypersurfaces about the complex hyperquadric \(Q\) and the real projective space \(\mathbb{R} P^n\) embedded in \(\mathbb{C} P^n\), in particular when \(P\) is totally geodesic. In the above mentioned paper they used the Jacobi equation technique while now they make use of the Riccati equation. Moreover, they derive comparisons for the relative volume, the first Dirichlet eigenvalue and the mean exit time. They also obtain several necessary geometrical properties when the equality is obtained in their comparison formulas but they are still unable to decide whether these equalities characterize the model pairs \((Q, \mathbb{C} P^n)\), \((\mathbb{R} P^n, \mathbb{C} P^n)\).
scientific article; zbMATH DE number 856556