arXiv research

We classify closed, simply-connected cohomogeneity-one Alexandrov spaces in dimensions $5$, $6$ and $7$. We show that every closed, simply-connected smooth $n$-orbifold, $2\leq n\leq 7$ with a cohomogeneity one action is equivariantly homeomorphic to a smooth good orbifold of cohomogeneity one.

Cohomogeneity-one actions on symmetric spaces of mixed type

Constructs harmonic maps between special geometric shapes.

We study isometric cohomogeneity one actions on the (n+1)-dimensional Minkowski space up to orbit-equivalence. We give examples of isometric cohomogeneity one actions on the Minkowski space whose orbit spaces are non-Hausdorff. We show that there exist isometric cohomogeneity one actions on the Minkowski space which ar…

An isometric action of a Lie group on a Riemannian manifold is of cohomogeneity one if the corresponding orbit space is one-dimensional. In this article we develop a conceptual approach to the classification of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type in terms of orbit equivalence. As…

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary, we obtain the classification of closed, $n$-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric $T^{n-1}$ action. In contra…

Study cohomogeneity one RCD-spaces, proving structural results and constructing new examples.

New structural result classifies actions on symmetric spaces.

Complete classification of symmetric spaces actions.

Abstract: Characterizes spaces with positive scalar curvature.

Constructs biharmonic and $r$-harmonic submanifolds in cohomogeneity one manifolds.

Classification results are given for (i) compact quaternionic Kähler manifolds with a cohomogeneity-one action of a semi-simple group, (ii) certain complete hyperKähler manifolds with a cohomogeneity-two action of a semi-simple group preserving each complex structure, (iii) compact 3-Sasakian manifolds which are cohomo…

No Einstein metrics found on certain double disk bundles.

Study finds metrics with positive intermediate Ricci curvature on specific low-dimensional manifolds.

In this paper, we provide a complete classification for all the isometric cohomogeneity one actions on unit spheres. Using this theory, we can very easily classify all the isometric cohomogeneity one actions on the Riemannian symmetric spaces $\mathbb{C}\mathrm{P}^{m-1}$ and $\mathbb{H}\mathrm{P}^{m-1}$.

Study cohomogeneity one expanding Ricci solitons on specific topologies.

Investigates harmonic self-maps' stability on cohomogeneity one manifolds.

We show that a certain family of cohomogeneity one manifolds does not admit an invariant metric of nonnegative sectional curvature, unless it admits one with positive curvature. As a consequence, the classification of nonnegatively curved cohomogeneity one manifolds in dimension 7 is reduced to only one further family …

The aim of this paper is to study cohomogeneity one isometric linear actions on the $p+q$-dimensional pseudo-Euclidean space $\mathbb{R}^{p,q}$. It is proved that the natural isometric action of the nilpotent factor of an Iwasawa decomposition of $SO(p,q)$ is not of cohomogeneity one. The orbits of cohomogeneity one ac…

The aim of this paper is to classify the cohomogeneity one conformal actions on the 3-dimensional Einstein universe $Ein^{1;2}$ , up to orbit equivalence. In a recent paper [21], we studied the unique (up to conjugacy) irreducible action of $PSL(2; \mathbb{R})$ on $Ein^{1;2}$ and we showed that the action is of cohomog…

Paper finds conditions for special geometric structures on certain spaces.

The Ambrose-Singer theorem is extended to cohomogeneity one Riemannian manifolds.

We prove a structure theorem for closed topological manifolds of cohomogeneity one; this result corrects an oversight in the literature. We complete the equivariant classification of closed, simply connected cohomogeneity one topological manifolds in dimensions $5$, $6$, and $7$ and obtain topological characterizations…

Study cohomogeneity-one Lagrangian mean curvature flow in complex spaces.

Classifies manifolds with quasipositive curvature.

We compute the rational Borel equivariant cohomology ring of a cohomogeneity-one action of a compact Lie group.

In this paper we give a full diffeomorphism characterization of compact simply connected cohomogeneity one manifolds in dimension six.

We classify simply connected, closed cohomogeneity one manifolds with singly generated or 4-periodic rational cohomology and positive Euler characteristic.

We analyse some properties of the cohomogeneity one Ricci soliton equations, and use Ansatze of cohomogeneity one type to produce new explicit examples of complete Kahler Ricci solitons of expanding, steady and shrinking types. These solitons are foliated by hypersurfaces which are circle bundles over a product of Fano…

Study on Hitchin index for cohomogeneity one nearly Kähler structures.

We classify those manifolds of positive euler characteristic on which a lie group G acts with cohomogeneity one, where G is classical simple

In this paper we classify, up to orbit equivalence, cohomogeneity one actions of connected closed Lie subgroups of $U(1,n)$ on the $(2n+1)$-dimensional anti de Sitter spacetime $AdS^{2n+1}$. We also give some new examples of nonproper cohomogeneity one actions on $AdS^{n+1}$ and determine parabolic Lie subgroups of $SO…

Proves Penrose inequality for cohomogeneity one initial data sets.

We show that if $G\times M \to M$ is a cohomogeneity one action of a compact connected Lie group $G$ on a compact connected manifold $M$ then $H^*_G(M)$ is a Cohen-Macaulay module over $H^*(BG)$. Moreover, this module is free if and only if the rank of at least one isotropy group is equal to the rank of $G$. We deduce …

We construct examples of cohomogeneity one special Lagrangian submanifolds in the cotangent bundle over the complex projective space, whose Calabi-Yau structure was given by Stenzel. For each example, we describe the condition of special Lagrangian as an ordinary differential equation. Our method is based on a moment m…

Classifies Einstein metrics on 4-manifolds with specific symmetry groups.

The aim of this paper is to classify the cohomogeneity one conformal actions on the three-dimensional essential Riemannian spaces, up to orbit equivalence. Among other results, the representations of all connected Lie groups acting with cohomogeneity one or zero within the full conformal group of a given three-dimensio…

Solves initial value problem for harmonic maps on specific manifolds.

We study gradient Ricci solitons with maximal symmetry. First we show that there are no non-trivial homogeneous gradient Ricci solitons. Thus the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. However, we apply the…

In this paper we give a characterization of the possible homology groups that can occur for compact simply connected cohomogeneity one manifolds in dimensions seven and lower.

New gradient Ricci solitons found for $SU(2)$ invariants.

New examples of hypersurfaces found in quaternionic hyperbolic spaces.

We classify, up to orbit equivalence, the cohomogeneity one actions on the noncompact duals of the symmetric spaces G_2, SU_3 and the real oriented two-plane Grassmannians.

Study on special Hermitian metrics on cohomogeneity one manifolds.

The study classifies manifolds with specific rational cohomology properties.

A cohomogeneity one manifold is a manifold with the action of a compact Lie group, whose quotient is one dimensional. Such manifolds are of interest in Riemannian geometry, in the context of nonnegative sectional curvature, as well as in other areas of geometry and in physics. In this paper we classify compact simply c…

Proves Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.

We classify superpotentials for cohomogeneity one Ricci solitons, finding new ones and proving non-existence in higher dimensions.