arXiv research

Study Mabuchi metrics on Fano manifolds proving their existence and properness.

Paper extends Calabi's extremal metric existence to compact Kähler manifolds.

The paper defines a new metric and studies proper biharmonic maps on tangent bundles.

Let $(X,ω)$ be a compact normal Kähler space, with Hodge metric $ω$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical Kähler metrics, we introduce a geodesic metric structure on $\mathcal H_ω(X)$, the space of Kähler potentials, whose completion is the fin…

The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.

Extends geometric group theory techniques to arbitrary proper metric ARs.

New proof of Kobayashi's properness criterion using metric geometry.

We compute the asymptotic dimension of the rationals given with an invariant proper metric. Also, we show that a countable torsion abelian group taken with an invariant proper metric has asymptotic dimension zero.

We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadra…

In this paper, we study geometric properties of quotient spaces of proper Lie groupoids. First, we construct a natural stratification on such spaces using an extension of the slice theorem for proper Lie groupoids of Weinstein and Zung. Next, we show the existence of an appropriate metric on the groupoid which gives th…

Let $(X,ω)$ be a compact Kähler manifold and $\mathcal H$ the space of Kähler metrics cohomologous to $ω$. If a cscK metric exists in $\mathcal H$, we show that all finite energy minimizers of the extended K-energy are smooth cscK metrics, partially confirming a conjecture of Y.A. Rubinstein and the second author. As a…

Proper holomorphic isometries between Bergman domains are biholomorphisms.

In this paper, we show that the existence of Sasakian-Einstein metrics is closely related to the properness of corresponding energy functionals. Under the condition that admitting no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of Sasakian-Einstein metric implies a Moser-Trudinger type i…

Metric WPD for pseudo-Anosov maps shows many unbounded quasi-morphisms.

We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rigidity theorems for Lie groupoids, which unify, simplify and improve si…

Study proper holomorphic maps between symmetric domains, proving rigidity and isometric properties.

We introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allow us to establish a Linearization Theorem for Ri…

The paper shows how contracting elements in groups lead to large quotients with specific growth rates.

Curve shortening flow is not unique on certain metrics.

Solves area minimizing surface problem in metric spaces with bounded genus.

Uniformly finite Cannon--Thurston fibers in most hyperbolic settings.

In this paper, we give a result on the properness of the K-energy, which answers a question of Song-Weinkove in any dimensions. Moreover, we extend our previous result on the properness of K-energy to the case of modified K-energy associated to extremal Kahler metrics.

The paper resolves a problem about metric inequivalence and characterizes proper holomorphic maps.

In this paper we prove that every proper Lie groupoid admits a desingularization to a regular proper Lie groupoid. When equipped with a Riemannian metric, we show that it admits a desingularization to a regular Riemannian proper Lie groupoid, arbitrarily close to the original one in the Gromov-Hausdorff distance betwee…

Proves the bending map is proper for hyperbolic 3-manifolds.

Critiques binary classification evaluation methods, advocating for proper scoring rules.

The paper proves uniqueness and existence of Kähler-Einstein metrics on certain compactifications.

New boundary for geodesic spaces captures Poisson boundary of mapping class groups.

Let Σbe a compact surface of type (g, n), n > 0, obtained by removing n disjoint disks from a closed surface of genus g. Assuming χ(Σ)<0, we show that on Σ, the set of flat metrics which have the same Laplacian spectrum of Dirichlet boundary condition is compact in the C^\infty topology. This isospectral compactness ex…

Proper actions on bornological spaces are characterized with compatible coarse structures.

In this paper, we prove the equivalence of the existence of extremal Kahler metrics and the properness of the modified K energy on projective bundles. Moreover, we discuss the relations of the lower boundedness of the K energy, the infimum of the Calabi energy and the extremal polynomials. In particular, we give an exa…

Well-known conjectures of Tian predict that existence of canonical Kahler metrics should be equivalent to various notions of properness of Mabuchi's K-energy functional. In some instances this has been verified, especially under restrictive assumptions on the automorphism group. We provide counterexamples to the origin…

We describe an explicit metric that induces the Chabauty topology on the space of closed subsets of a proper metric space M.

We study the mechanisms of the non properness of the action of the group of diffeomorphisms on the space of Lorentzian metrics of a compact manifold. In particular, we prove that nonproperness entails the presence of lightlike geodesic foliations of codimension 1. On the 2-torus, we prove that a metric with constant cu…

In this paper, we apply the method developed in [Ti97] and [TZ00] to proving the properness of log $F$-functional on any conic Kähler-Einstein manifolds. As an application, we give an alternative proof for the openness of the continuity method through conic Kähler-Einstein metrics.

Maximal acceleration metrics limit spacetime curvature.

This paper argues against using calibration metrics for assessing posterior probabilities and proposes expected proper scoring rules instead.

Riemannian metrics on orbifolds are equivalent to diffeological ones.

Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not amenable then its second continuous bounded cohomology group with coefficients the regular representation does not vanish. This yields some structure results for such groups.

The present paper deals with the study of generalized $φ$-recurrent generalized $(k,μ)$-contact metric manifolds with the existence of such notion by a proper example.

Tian's conjectures solved in Kahler geometry, linking metrics and inequalities.

New groups found that are similar but not the same in terms of geometry.

New examples of weakly Einstein conformal products are constructed.

Bornological metrics on groups are studied, showing equivalence classes and constructing non-equivalent improper metrics.

The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (sigma-immanent description). Constructing the geometry, one does not use topology and topological properties. For instance, the straight, passing through points A and B, is defined as a set …

Paper proves existence of constant scalar curvature Kähler metrics under certain conditions.

Decision-alignment evaluates uncertainty quantification for decision-relevant UQ

The paper studies geometric properties of higher genera for proper Lie group actions.