Isoparametric foliations on complex projective spaces

Irreducible isoparametric foliations of arbitrary codimension q on complex projective spaces CP^n are classified, except if n=15 and q=1. Remarkably, there are noncongruent examples that pull back under the Hopf map to congruent foliations on the sphere. Moreover, there exist many inhomogeneous isoparametric foliations, even of higher codimension. In fact, every irreducible isoparametric foliation on the complex projective n-space is homogeneous if and only if n+1 is prime. The main tool developed in this work is a method to study singular Riemannian foliations with closed leaves on complex projective spaces. This method is based on certain graph that generalizes extended Vogan diagrams of inner symmetric spaces.Comment: 39 pages, minor revision, to appear in Trans. Amer. Math. So

Polar foliations on quaternionic projective spaces

We classify irreducible polar foliations of codimension $q$ on quaternionic projective spaces $\mathbb H P^n$, for all $(n,q)\neq(7,1)$. We prove that all irreducible polar foliations of any codimension (resp. of codimension one) on $\mathbb H P^n$ are homogeneous if and only if $n+1$ is a prime number (resp. $n$ is even or $n=1$). This shows the existence of inhomogeneous examples of codimension one and higher

Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms

We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do not exist. In complex hyperbolic spaces these are holomorphically congruent to open parts of tubes around the ruled minimal submanifolds with totally real normal bundle introduced by Berndt and Bruck. In particular, they are open parts of homogenous ones

Isoparametric hypersurfaces in Damek-Ricci spaces

We construct uncountably many isoparametric families of hypersurfaces in Damek-Ricci spaces. We characterize those of them that have constant principal curvatures by means of the new concept of generalized Kahler angle. It follows that, in general, these examples are inhomogeneous and have nonconstant principal curvatures. We also find new cohomogeneity one actions on quaternionic hyperbolic spaces, and an isoparametric family of inhomogeneous hypersurfaces with constant principal curvatures in the Cayley hyperbolic plane.Comment: Some references update

Inhomogeneous isoparametric hypersurfaces in complex hyperbolic spaces

We construct examples of inhomogeneous isoparametric real hypersurfaces in complex hyperbolic spaces

Solutions to the overdetermined boundary problem for semilinear equations with position-dependent nonlinearities

We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the Euclidean space and on compact Riemannian manifolds. As a byproduct of the proofs we also obtain some rigidity, or partial symmetry, results for solutions to overdetermined problems on Riemannian manifolds of nonconstant curvature.Comment: 36 page

Canonical extension of submanifolds and foliations in noncompact symmetric spaces

We propose a method to extend submanifolds, singular Riemannian foliations and isometric actions from a boundary component of a noncompact symmetric space to the whole space. This extension method preserves minimal submanifolds, isoparametric foliations and polar actions, among other properties. One of the several applications yields the first examples of inhomogeneous isoparametric hypersurfaces in noncompact symmetric spaces of rank at least two.Comment: 9 page

On homogeneous manifolds whose isotropy actions are polar

We show that simply connected Riemannian homogeneous spaces of compact semisimple Lie groups with polar isotropy actions are symmetric, generalizing results of Fabio Podesta and the third named author. Without assuming compactness, we give a classification of Riemannian homogeneous spaces of semisimple Lie groups whose linear isotropy representations are polar. We show for various such spaces that they do not have polar isotropy actions. Moreover, we prove that Heisenberg groups and non-symmetric Damek-Ricci spaces have non-polar isotropy actions.Comment: 20 page

Isoparametric hypersurfaces in complex hyperbolic spaces

We classify isoparametric hypersurfaces in complex hyperbolic spaces.Comment: Minor changes, 46 page

Isoparametric submanifolds in two-dimensional complex space forms

We show that an isoparametric submanifold of a complex hyperbolic plane, according to the definition of Heintze, Liu and Olmos', is an open part of a principal orbit of a polar action. We also show that there exists a non-isoparametric submanifold of the complex hyperbolic plane that is isoparametric according to the definition of Terng's. Finally, we classify Terng-isoparametric submanifolds of two-dimensional complex space forms