21 problems

Generalized Alekseevskii conjecture. Every expanding homogeneous Ricci soliton is isometric to a simply-connected solvmanifold.

Standardness conjecture. Solvmanifolds exhaust the class of noncompact homogeneous Einstein manifolds.

Let $G$ be a simple Lie group and let $P$ be a parabolic subgroup of $G$. A domain in $G/P$ is proper almost-homogeneous if it is proper and admits an automorphism group with a com…

Let $G/P$ be a flag manifold, where $G/P$ is neither real projective space nor the conformal sphere. A proper divisible domain is a proper domain in $G/P$ admitting a properly disc…

Let $g_{(a,b,c)}$ be the homogeneous metric on the Lie group $\operatorname{SU}(2)$ or $\operatorname{SO}(3)$ determined by positive parameters $a,b,c$. For the Hodge Laplacian…

Let $M=\mathsf G/\mathsf H$ be one of the low-dimensional full flag manifolds listed in Theorem, with $\mathsf G$ a compact simple Lie group of type ${\rm A}_n$, ${\rm B}_n$,…

Let $M=\mathsf G/\mathsf H$ be a compact homogeneous space whose isotropy representation consists of pairwise inequivalent irreducible summands. In particular, this includes the ca…

A homogeneous Einstein manifold is a Riemannian manifold with a transitive isometry group and a metric satisfying $\operatorname{Ric}_g=\lambda g$ for a constant $\lambda$. Aleksee…

Non-solvsoliton conjecture. If the one-loop deformation has $c>0$ and dimension $4n>4$, then every hypersurface orbit of its cohomogeneity-one action, with the induced metric, is n…

Let $g_1,\dots,g_k$ be a family of positively related Riemannian metrics on a homogeneous space. A family is positively related when its metrics admit a common decomposition … with…

Homogeneous harmonic-curvature conjecture. Any homogeneous Riemannian manifold $M$ with harmonic curvature is Ricci-parallel:

For $\alpha\in(0,1]$, define … Here $b_{x_0,\alpha}$ is the coordinate function and $P(x_0,\alpha)$ is the associated period. The midpoint-coordinate limit conjecture. $L(\alpha)$…

For $n$ in the range considered by the complex Stiefel manifolds, let $V_2\mathbb{C}^{n+2}=\operatorname{SU}(n+2)/\operatorname{SU}(n)$ be the complex Stiefel manifold of orthonorm…

Let $n\geq 13$. Consider the compact Lie group $\operatorname{SU}(n+3)$ and left-invariant Einstein metrics that are non naturally reductive. The authors' conjecture. For every…

Let $P$ be a complex Lie group and let $\boldsymbol{\pi}$ be a discrete cocompact subgroup, so that $P/\pi$ is a compact quotient. Invariant-structure conjecture. Every holomorphic…

An expanding homogeneous Ricci soliton is a homogeneous Ricci soliton with soliton constant $\lambda<0$. A solvmanifold is a homogeneous space associated with a solvable Lie group;…

An expanding algebraic soliton is a homogeneous Ricci soliton whose soliton structure is algebraic. Euclidean-space conjecture. Any expanding algebraic soliton is diffeomorphic to…

A homogeneous manifold is a manifold equipped with a transitive group action by isometries, and an Einstein manifold has Ricci tensor proportional to its metric. Its scalar curvatu…

Let $p,q,n$ be as in the two homogeneous pairs $(O(p+2,q),O(p,q))$ and $(O(p+1,q+1),O(p,q))$, let $t$ be the parameter, and let $(z_1,\ldots,z_n,w)$ be coordinates on…

Let $M=G/H$ be a homogeneous Ricci soliton with $c<0$, where $G$ is a transitive Lie group and $H$ is the isotropy subgroup. A homogeneous Ricci soliton is a Riemannian homogeneous…

Let $M$ be a noncompact, nonflat, homogeneous Einstein manifold, and let $G$ be its isometry group. An isotropy subgroup is the stabilizer in $G$ of a point of $M$; a maximal compa…