arXiv research

The paper studies proper discontinuity of actions on Weyl chamber flow spaces.

New resonance theory for Anosov flows connects spectral properties to mixing measures.

We consider the family of harmonic measures on a lamination $\mathcal{L}$ of a compact space $X$ by locally symmetric spaces $L$ of noncompact type, i.e. $L\simeq Γ_L\backslash G/K$. We establish a natural bijection between these measures and the measures on an associated lamination foliated by $G$-orbits, $\hat{\mathc…

New compactification for character varieties with good topological properties.

In this paper we present a topological way of building a compactification of a symmetric space from a compactification of a Weyl Chamber.

This paper begins with an observation that the isospectral leaves of the signed Toda lattice as well as the Toda flow itself may be constructed from the Tomei manifolds by cutting and pasting along certain chamber walls inside a polytope. It is also observed through examples that although there is some freedom in this …

We investigate discrete groups $G$ of isometries of a complete connected Riemannian manifold $M$ which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the the orbit space $M/G$ is isometric to a Weyl chamber $C$ which is a Riemannian …

Study SRB measures for Anosov actions on manifolds.

We study the Weyl chamber length boundary both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group. This result is obtained as consequence of a canonical decomposition of a geodesic current on a surface of finite type arising…

Develops Poisson and Dirac manifolds of compact types with applications.

Study convex hulls of orbits for compact groups, defining new invariants related to polynomial degrees.

In a symmetric space of noncompact type X = G/K oriented geodesic segments correspond to points in the Euclidean Weyl chamber. We can hence assign vector-valued side-lengths to segments. Our main result is a system of homogeneous linear inequalities describing the restrictions on the side -lengths of closed polygons. T…

We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kahler manifolds and show the necessity of the Kahler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying…

This paper connects real closed fields to Hitchin representations and their properties.

Study pinches curvature under Laplacian G_2 flow, proving Weyl tensor norm blows up.

Consider a Hamiltonian action of a compact Lie group on a compact symplectic manifold. A theorem of Kirwan's says that the image of the momentum mapping intersects the positive Weyl chamber in a convex polytope. I present a new proof of Kirwan's theorem, which gives explicit information on how the vertices of the polyt…

We compute the evolution equation of the Weyl tensor under the Ricci flow of a Riemannian manifold and we discuss some consequences for the classification of locally conformally flat Ricci solitons.

We introduce a basis of the Orlik-Solomon algebra labeled by chambers, so called chamber basis. We consider structure constants of the Orlik-Solomon algebra with respect to the chamber basis and prove that these structure constants recover D. Cohen's minimal complex from the Aomoto complex.

In this paper, we first derive a pinching estimate on the traceless Ricci curvature in term of scalar curvature and Weyl tensor under the Ricci flow. Then we apply this estimate to study finite-time singularity behavior. We show that if the scalar curvature is uniformly bounded, then the Weyl tensor has to blow up, as …

In this paper we will extend to non-abelian groups inverse spectral results, proved by us in an earlier paper, for compact abelian groups, i.e. tori. More precisely, Let $\mathsf G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $X$. We will show that for the Schrödinger operator $-\hbar^2…

Develops a new calculus for studying operators on principal bundles.

Let $Γ$ be a finitely generated group and $G$ be a noncompact semisimple connected real Lie group with finite center. We consider the space $\mathcal X$ of conjugacy classes of reductive representations of $Γ$ into $G$. We define the {\it translation vector} of an element $g$ in $G$, with values in a Weyl chamber, as a…

A generalized cusp $C$ is diffeomorphic to $[0,\infty)$ times a closed Euclidean manifold. Geometrically $C$ is the quotient of a properly convex domain by a lattice, $Γ$, in one of a family of affine groups $G(ψ)$, parameterized by a point $ψ$ in the (dual closed) Weyl chamber for $SL(n+1,\mathbb{R})$, and $Γ$ determi…

Characterizes solutions to Z-critical equations on surfaces using effective conditions.

Study of Bach flow on specific nilmanifolds, converging to a soliton.

On a cotangent bundle $T\sp*G$ of a Lie group $G$ one can describe the standard Liouville form $θ$ and the symplectic form $d θ$ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of $G$ on itself), and also in terms of the left Maurer-Cartan form and the right moment mapping, and…

In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than one and their geometric perturbations. Our goal here is to explain how analysis of the Laplacian on the globally symmetric space $\SL(3,\RR)/\…

We investigate Bartnik's static metric extension conjecture under the additional assumption of axisymmetry of both the given Bartnik data and the desired static extensions. To do so, we suggest a geometric flow approach, coupled to the Weyl-Papapetrou formalism for axisymmetric static solutions to the Einstein vacuum e…

Anosov groups in rank ≤3 have unique ergodic horospherical actions.

As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber; in addition to the metric length it contains information on the direction of the segment. We study in this paper restrictions on the vector valued s…

Study on Ricci flow on 4-spheres, proving standard sphere convergence.

We prove pinching estimates for solutions of the linearized Ricci flow system on a closed manifold of dimension $n\geq 4$ with positive scalar curvature and vanishing Weyl tensor. If the vanishing Weyl tensor condition is removed, we only give a rough pinching estimate controlled by some blow-up function in a short tim…

New proof shows perturbed non-compact Einstein spaces attract to unique global solution.

In this paper the rate relations of Riemann, conformal, conharmonic and Weyl curvature tensors under Yamabe flow are studied. Modified Riemann extensions under Yamabe flow is discussed. The paper ends with remarks on some standard metrics.

Developed causal chambers for AI validation, providing real-world data.

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the Abelian vortex equations, and it is shown that the property that a metric solve t…

The paper studies chambered invariants of real Cauchy-Riemann operators on Riemann surfaces.

Modular curves $X_{1}(N)$ parametrize elliptic curves with a point of order $N$. They can be identified with connected components of projectivized strata $\mathbb{P}\mathcal{H}(a,-a)$ of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the …

Study compactifies representations space of hyperbolic surfaces.

Projections from flats to maximal flats defined and studied.

We establish a $C^{1,α}$ compactness theorem for the metrics with bounded self - dual Weyl tensor and Scalar curvature. The key step is to estimate the $C^{1,α}$ harmonic radius, where we use the blow up analysis as in \cite{Anderson90}. The result is motivated by, and may be applied to the Calabi flow on complex surfa…

In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result \cite{Cao2011} in the Ricci flow. In dimension four, we also study the integral bound of the "Riemann curvature" for the Ricci-harmonic flow generalizing a recently result of…

Study on curvature conditions for non-conformally flat spheres using quasiconformal maps and Ricci flow.

In this paper we set up the family Seiberg-Witten theory. It can be applied to the counting of nodal pseudo-holomorphic curves in a symplectic 4-manifold (especially a Kahler surface). A new feature in this theory is that the chamber structure plays a more prominent role. We derive some wall crossing formulas measuring…

We investigate the geometry in a real Euclidean building X of type A2 of some simple configurations in the associated projective plane at infinity P, seen as ideal configurations in X, and relate it with the projective invariants (from the cross ratio on P). In particular we establish a geometric classification of gene…

It is shown that 3D part of a spherically symmetric solution in conformal Weyl gravity interacting with Maxwell electrodynamics is a Yamabe flow as well. The Yamabe flow describes the transition from a horn of an initial wormhole to a 3D Euclidean space both filled with a radial electric field. It is supposed that such…

Let $M$ be a cohomogeneity one manifold of a compact semisimple Lie group $G$ with one singular orbit $S_0 = G/H$. Then $M$ is $G$- diffeomorphic to the total space $G \times_H V$ of the homogeneous vector bundle over $S_0$ defined by a sphere transitive representation of $G$ in a vector space $V$. We describe all such…

Theory developed for Hilbert geometry over valued fields, linking real and non-Archimedean geometries.