arXiv research

We explain how the generalized Milnor-Wood inequality for reductive representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie group translates, under the non-abelian Hodge correspondence, into various kinds of Milnor-Wood inequalities for Higgs bundles. This clarifies the relation between the repre…

New proof of Milnor-Wood inequality for circle bundles.

Generalizes Toledo invariant to singular klt varieties, proving Milnor-Wood inequality.

Paper uses index theorem to relate symplectic bundle signature to surface group representation in real symplectic group.

We prove a Milnor-Wood inequality for representations of the fundamental group of a compact complex hyperbolic manifold in the group of isometries of quaternionic hyperbolic space. Of special interest is the case of equality, and its application to rigidity. We show that equality can only be achieved for totally geodes…

This note describes sharp Milnor--Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern--Sullivan's conjecture in this case. The…

We introduce a notion of the twist of an isometry of the hyperbolic plane. This twist function is defined on the universal covering group of orientation-preserving isometries of the hyperbolic plane, at each point in the plane. We relate this function to a function defined by Milnor and generalised by Wood. We deduce v…

Characterizes components of representations space for punctured surfaces.

We prove an extension of Milnor-Wood inequalities to a geometric situation. We study representations of the fundamental group of a compact manifold into the isometry group of a product of rank one spaces of the same dimension and show an upper bound on the volume of the representation. When the target group is the isom…

We define the Toledo invariant of a G-Higgs bundle on a Riemann surface, where G is a real semisimple group of Hermitian type, and we prove a Milnor-Wood type bound for this invariant when the bundle is semistable. We prove rigidity results when the Toledo invariant is maximal, establishing in particular a Cayley corre…

We propose a definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple Lie groups of Hermitian type. This definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting: …

For a based manifold (M,*), the question of whether the surjection Diff(M,*) \rightarrow π_0 Diff(M,*) admits a section is an example of a Nielsen realization problem. This question is related to a question about flat connections on M-bundles and is meaningful for M of any dimension. In dimension 2, Bestvina-Church-Sou…

As for the theory of maximal representations, we introduce the volume of a Zimmer's cocycle $Γ\times X \rightarrow \mbox{PO}^\circ(n, 1)$, where $Γ$ is a torsion-free (non-)uniform lattice in $\mbox{PO}^\circ(n, 1)$, with $n \geq 3$, and $X$ is a suitable standard Borel probability $Γ$-space. Our numerical invariant ex…

We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old conjecture proved by Milnor in dimension 2. In particular, the Chern conjecture follows…

We complete the classification of maximal representations of uniform complex hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional groups ${\rm E}_6$ and ${\rm E}_7$. We prove that if $ρ$ is a maximal representation of a uniform complex hyperbolic lattice $Γ\subset{\rm SU}(1,n)$, $n>1$, in an exce…

We investigate representations of Kähler groups $Γ= π_1(X)$ to a semisimple non-compact Hermitian Lie group $G$ that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor--Wood inequality similar to those found by Burger--Iozzi and Koziarz--Maubon. Thanks…

Let G be either SU(p,2) with p>=2, Sp(2,R) or SO(p,2) with p>=3. The symmetric spaces associated to these G's are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kähler manifolds and Higgs b…

The paper refines Mather-Thurston theorems for flat connections in manifolds.

In this paper we study the moduli space of representations of a surface group (i.e., the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n,R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor-Wood type inequality…

Researchers calculate the volume of Seifert representations for graph manifolds and their covers.

The purpose of this article is two-fold: We first give a more elementary proof of a recent theorem of Korkmaz, Monden, and the author, which states that the commutator length of the n-th power of a Dehn twist along a boundary parallel curve on a surface with boundary S of genus g at least two is the floor of (|n|+3)/2 …

We prove the existence of foliations transverse to pseudo-Anosov flows using veering triangulations.

Using the L^2 norm of the Higgs field as a Morse function, we study the moduli spaces of U(p,q)-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on (p,q). A key step is the identification of the function's local minima as moduli spaces of holomorph…

In this paper, we study the global behaviour of contact structures on oriented manifolds V which are circle bundles over a closed orientable surface S of genus g>0. We establish in particular contact analogs of a number of classical results about foliations due to Milnor, Wood, Thurston, Matsumoto, and Ghys. In Section…

The isoperimetric inequality and related inequalities are explored.

New proof of Willmore inequality using geometric divergence inequality.

Lorentz-Finsler geometry reveals new and old inequalities.

The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.

The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.

New inequality on sphere generalizes circle inequality.

Paper proves anisotropic Minkowski inequality and related inequalities.

Explains geometric inequalities for minimal hypersurfaces.

The paper finds new inequalities for convex polygons.

The study improves Bochner inequality on Finsler manifolds to derive important inequalities.

The paper proves various inequalities on gradient shrinking Ricci solitons.

Sharp inequality found on three-balls for fourth order Sobolev traces.

Sharp inequalities for star bodies in 2D space.

The paper develops inequalities for log-concave functions and related surface areas.

Study on functional inequalities on simple edge spaces.

Proves inequalities on curved spaces with positive curvature.

The paper derives inequalities on Finsler manifolds, influenced by their curvatures.

Paper refines Talagrand inequality on Euclidean spaces.

New inequalities for convex curves with multiple geometric factors.

Alternative proofs for various inequalities on Riemannian manifolds.

The paper establishes inequalities and gradient estimates for harmonic functions on Finsler measure spaces.

The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.

The paper explores how information geometry impacts classical CR inequalities.

Extends Riemannian geometry inequalities with sharper estimates.