arXiv research

New technique shows non-orbifold singularities are undetectable by G-invariant spectrum.

Given a compact boundaryless Riemannian manifold $Y$ on which a compact Lie group $G$ acts, there is always a metric on $Y$ such that the action is by isometries. Assuming $Y$ is equipped with such a metric, recall that the $G$-invariant Laplacian is the restriction of the ordinary Laplacian to the space of functions w…

The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere $S^2$ endowed with $S^1$-invariant metrics, we consider the subsequence $λ_k^G$ of the spectrum of a Riemannian manifold $M$ which corresponds…

Formula for Lichnerowicz Laplacian on invariant metrics, deducing stability of Einstein manifolds.

Let X=G/K be a symmetric space of noncompact type and let L be the Laplacian associated with a G-invariant metric on X. We show that the resolvent kernel of L admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane w…

We study quantum moment maps of $G$-invariant star products, which are a quantum analogue of the moment map for classical Hamiltonian systems. Introducing an integral representation, we show that any quantum moment map for a $G$-invariant star product is differentiable. This property gives us a new method for the class…

Study of cubulations in hyperbolic groups and their invariant cross ratios.

Let O be a nilpotent orbit in g^C where G is a compact, simple group and g=Lie(G). It is known that O carries a unique G-invariant hyperKähler metric admitting a hyperKähler potential compatible with the Kirillov-Kostant-Souriau symplectic form. In this work, the hyperKähler potential is explicitly calculated when O is…

We consider the equivariant Yamabe problem, i.e. the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their …

Conditions for pre-quantizability of G-invariant forms are derived using moment maps.

This paper classifies $G$-invariant shallow neural networks.

Efficient neural network invariant to symmetry subgroups.

The study proves a generic multiplicity one theorem for $G$-invariant minimal hypersurfaces.

We show a correspondence between the set of all G-invariant projectively flat connections on a homogeneous apace $M=G/K$, and the one of all {G}^~-invariant flat connections on a homogeneous space {M}^~={G}^~/K, where {G}^~ is a central extension of G.

A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of …

Compact Lie groups can be realized as automorphism groups of Riemannian manifolds.

Ricci flow on certain homogeneous spaces creates metrics with positive curvature.

Infinite $G$-invariant minimal hypersurfaces found in Riemannian manifolds.

The paper proves the existence of $G$-invariant minimal hypersurfaces on certain Riemannian manifolds.

Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.

Study analyzes spectral properties on specific geometric spaces.

The paper studies Randers and $(α,β)$ equigeodesics on compact homogeneous manifolds.

Study proves existence of special surfaces in geometric spaces.

Study stability of Einstein metrics on homogeneous spaces.

Classifies reversible Finsler metrics with positive curvature.

Describes metrics on homogeneous spaces with equivalent isotropy summands.

We show that Cheeger deformations regularize $G$--invariant metrics in a very strong sense.

The paper generalizes free boundary min-max theory to equivariant settings.

Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow

New Kazdan-Warner problem for equivariant metrics on manifolds.

Incompressible fluid dynamics on special manifolds.

The paper studies invariant metrics with positive scalar curvature on 3-manifolds.

Analytic realization of Thom-Smale complex for G-manifolds.

Let $M$ be a Riemannian manifold with a polar action by the Lie group $G$, with section $Σ\subset M$ and generalized Weyl group $W$. We show that restriction to $Σ$ is a surjective map from the set of smooth $G$-invariant tensors on $M$ onto the set of smooth $W$-invariant tensors on $Σ$. Moreover, we show that every s…

Study shows precise Szegö kernel behavior for CR manifolds with group actions.

Researchers describe a unique hyper-Kähler structure on a specific domain.

We present a new construction of tubular neighborhoods in (possibly infinite dimensional) Riemannian manifolds M, which allows us to show that if G is an arbitrary group acting isometrically on M, then every G-invariant submanifold with locally trivial normal bundle has a G-invariant total tubular neighborhood. We appl…

Study of invariant solutions for certain PDEs on Riemannian manifolds.

Study quasi-Einstein metrics on cohomogeneity-one manifolds.

Deep neural networks can approximate invariant/equivariant functions with fewer parameters.

Let $G$ be a compact connected Lie group and $H$ a closed subgroup of $G$. Suppose the homogeneous space $G/H$ is effective and has dimension 3 or higher. Consider a $G$-invariant, symmetric, positive-semidefinite, nonzero (0,2)-tensor field $T$ on $G/H$. Assume that $H$ is a maximal connected Lie subgroup of $G$. We p…

On a smooth closed oriented $4$-manifold $M$ with a smooth action of a finite group $G$ on a Spin$^c$ structure, $G$-monopole invariant is defined by "counting" $G$-invariant solutions of Seiberg-Witten equations for any $G$-invariant Riemannian metric on $M$. We compute $G$-monopole invariants on some $G$-manifolds. F…

In this note we classify all homogeneous spaces $G/H$ admitting a $G$-invariant $G_2$-structure, assuming that $G$ is a compact Lie group and $G$ acts effectively on $G/H$. They include a subclass of all homogeneous spaces $G/H$ with a $G$-invariant $\tilde G_2$-structure, where $G$ is a compact Lie group. There are ma…

Invariant Kähler metrics on line bundles are derived from the Calabi ansatz.

New research shows invariant networks can approximate any continuous function.

Researchers study metrics with maximal Ricci curvature on homogeneous spaces.

New neural network architectures use signed permutation representations for finite groups, improving performance.

In this paper, we characterize the dynamic of every abelian subgroups $\mathcal{G}$ of GL($n$, $\mathbb{K}$), $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. We show that there exists a $\mathcal{G}$-invariant, dense open set $U$ in $\mathbb{K}^{n}$ saturated by minimal orbits with $\mathbb{K}^{n}- U$ a union of at most $n$…