On submanifolds whose tubular hypersurfaces have constant mean curvatures

Motivated by the theory of isoparametric hypersurfaces, we study submanifolds whose tubular hypersurfaces have some constant "higher order mean curvatures". Here a $k$-th order mean curvature $Q_k$ ($k\geq1$) of a hypersurface $M^n$ is defined as the $k$-th power sum of the principal curvatures, or equivalently, of the shape operator. Many necessary restrictions involving principal curvatures, higher order mean curvatures and Jacobi operators on such submanifolds are obtained, which, among other things, generalize some classical results in the theory of isoparametric hypersurfaces given by E. Cartan, K. Nomizu, H. F. M{\"u}nzner, Q. M. Wang, \emph{etc.}. As an application, we finally get a geometrical filtration for the focal varieties of isoparametric functions on a complete Riemannian manifold.Comment: 29 page

Comparison theorems for manifolds with mean convex boundary

Let M-n be an n-dimensional Riemannian manifold with boundary partial derivative M. Assuming that Ricci curvature is bounded from below by (n - 1)k, for k is an element of R, we give a sharp estimate of the upper bound of rho(x) = d(x, partial derivative M), in terms of the mean curvature bound of the boundary. When partial derivative M is compact, the upper bound is achieved if and only if M is isometric to a disk in space form. A Kahler version of estimation is also proved. Moreover, we prove a Laplacian comparison theorem for distance function to the boundary of Kahler manifold and also estimate the first eigenvalue of the real Laplacian.SCI(E)[email protected]