arXiv research

We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we …

The study explores new metric structures on manifolds, linking them to Einstein metrics.

New structure with B-metric extends classical almost contact structures.

Study weak quasi contact metric manifolds to generalize K-contact and Sasakian manifolds criteria.

Study on Ricci solitons and Einstein metrics in weak β-Kenmotsu manifolds.

Study weak $f$-K-contact manifolds, finding Einstein-type metrics and solitons.

The paper finds the Finsler structure of Apollonian weak metric on unit disc.

The paper explores generalized Riemannian manifolds with weak metric structures.

Study $\ast$-$η$-Ricci solitons on weak Kenmotsu $f$-manifolds.

Study weak geodesic lines in Kähler metric space, disproving a conjecture.

Survey of weak metric f-manifolds, generalizing K. Yano's structures.

We define and study metrics and weak metrics on the Teichmueller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of properly embedded arcs in the surface. We give a comparison between the defined m…

The paper describes geometric properties of Teichmüller space metrics.

In this paper, we study general $(α,β)$-metrics which $α$ is a Riemannian metric and $β$ is an one-form. We have proven that every weak Landsberg general $(α,β)$-metric is a Berwald metric, where $β$ is a closed and conformal one-form. This show that there exist no generalized unicorn metric in this class of general $(…

Introduces a new framework for Riemannian diffeology.

New structures defined for studying contact foliations and their geometry.

Paper investigates conditions for independence of weak gradients on metric spaces.

New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.

The paper explores new connections in non-symmetrical gravitational theory using weak metric structures.

Study weak super Ricci flow through neckpinch in metric measure spaces.

We define the notion of weak Minkowski metric and prove some basic properties of such metrics. We also highlight some of the important analogies between Minkowski geometry and the Funk and Hilbert geometries.

Study on a new type of manifolds that generalize almost C-manifolds.

Study explores weak generalized K-contact structures in contact metric spaces.

David Hilbert discovered in 1895 an important metric that is canonically associated to any convex domain $Ω$ in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof assumes a certain degree of smoothness of the boundary of $Ω$ and refers to a theorem by Busemann and Mayer that…

Surveying Ricci flow for weak lower scalar curvature bounds.

In this paper, we discuss a Donaldson's version of the modified $K$-energy associated to the Calabi's extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified $K$-energy.

Study the geometry of weak para-f-structures and subclasses.

In the category of metrics with conical singularities along a smooth divisor with angle in $(0, 2π)$, we show that locally defined weak solutions ($C^{1,1}-$solutions) to the Kähler-Einstein equations actually possess maximum regularity, which means the metrics are actually Hölder continuous in the singular polar coord…

In this paper, we prove that on a Fano manifold $M$ which admits a Kähler-Ricci soliton $(\om,X)$, if the initial Kähler metric $\om_{\vphi_0}$ is close to $\om$ in some weak sense, then the weak Kähler-Ricci flow exists globally and converges in Cheeger-Gromov sense. Moreover, if $\vphi_0$ is also $K_X$-invariant, the…

The paper characterizes Sasakian manifolds using weak nearly Sasakian structures.

Study of $η$-Ricci solitons and $η$-Einstein metrics on weak $β$-Kenmotsu $f$-manifolds.

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

New method estimates model performance bounds without ground truth labels.

We prove that the L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L^2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponentia…

We study weakened $f$-structures on manifolds, generalizing classical results.

On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harm…

The existence of \emph{weak conical Kähler-Einstein} metrics along smooth hypersurfaces with angle between $0$ and $2π$ is obtained by studying a smooth continuity method and a \emph{local Moser's iteration} technique. In the case of negative and zero Ricci curvature, the $C^0$ estimate is unobstructed; while in the ca…

New formulations for Ricci flows without smoothness.

We study algebro-geometric consequences of the quantised extremal Kähler metrics, introduced in the previous work of the author. We prove that the existence of quantised extremal metrics implies weak relative Chow polystability. As a consequence, we obtain asymptotic weak relative Chow polystability and $K$-semistabili…

We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full $H^2$-metric without zero order terms. We find isometries (called $R$-transforms) from some of these spaces i…

Compactifies Calabi-Yau to weak Fano manifolds.

We consider some metrics and weak metrics defined on the Teichmueller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comp…

New ensemble models classify mouse movement trajectories to assess survey question difficulty.

We study the quantization of coupled Kähler-Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, so called the ``balanced metrics''. We prove the existence and weak convergence of balanced metrics for the negative first Chern class, while for the positive first Chern clas…

Study curvature properties of w.a. S-manifolds with new conditions.

We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kahler potentials on a compact Kahler manifold thus confirming a conjecture of Chen and give some applications in Kahler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally …

We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of $C^{1, 1}$-regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for $C^{1,1}$-metrics, an…

The continuity method is used to deform the cone angle of a weak conical Kähler-Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldson's Openness Theorem on deforming cone angle \cite{Don} by combining it with the regulari…