12 problems

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 a compact homogeneous space whose isotropy representation consists of pairwise inequivalent irreducible summands. In particular, this includes the ca…

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. $L(\alpha)$…

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 $M=G/H$ 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…