arXiv research

New examples of non-bumpy metrics on spheres and projective spaces with multiplicity.

In this paper, we prove that the $3$-sphere endowed with an arbitrary Riemannian metric either contains at least two embedded minimal $2$-spheres or admits an optimal foliation by $2$-spheres. This generalizes recent results by Haslhofer-Ketover (Duke Math. J. 2019), where the existence of optimal foliations and minima…

Intuition drawn from quantum mechanics and geometric optics raises the following long-standing question: can the length spectrum of a closed Riemannian manifold be recovered from its Laplace spectrum? The Poisson relation states that for any closed Riemannian manifold $(M,g)$ the singular support of the trace of its wa…

This thesis surveys various metrics on Riemann surface spaces.

Proves existence and uniqueness of weighted metrics for smooth spaces.

New Finsler metrics constructed from $GDW$-metrics.

In this essay, we study the sufficient and necessary conditions for a Randers metrc to be of constant Ricci curvature without the restriction of strong convexity (regularity). The classification result for the case $\|β\|_α>1$ is provided, which is similar to the famous Bao-Robles-Shen's result for strongly convex Rand…

We prove the equivalences of several classical complete metrics on the Teichmüller and the moduli spaces of Riemann surfaces. We use as bridge two new Kähler metrics, the Ricci metric and the perturbed Ricci metric and prove that the perturbed Ricci metric is a complete Kähler metric with bounded negative holomorphic s…

New metric defined for bounded symmetric domains.

Introduces Finslerian convolution metrics and their properties.

Survey of recent metric geometry in Kähler metrics space.

Study on geodesics of Finsler metrics derived from Riemannian metrics.

We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L^2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi's metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geome…

New Kähler metrics generalize Calabi's and relate to Fano manifolds.

We study the geodesic equation for the Dirichlet (gradient) metric in the space of Kaehler potentials. We first solve the initial value problem for the geodesic equation of the combination metric, including the gradient metric. We then discuss a comparison theorem between it and the Calabi metric. As geometric motivati…

New Finsler metrics defined by Riemannian and 1-forms are studied.

Sharp estimates for Finsler metrics in convex domains.

Study on special Finsler metrics with conditions for Riemannian and isotropic properties.

The study examines Lee metrics on groups and their properties.

Survey of spectral, probabilistic, and deep metric learning methods.

Study shows convergence of Lagrangian submanifolds under certain metrics.

In this paper, we study an important class of Finsler metrics--square metrics. We give two expressions of such metrics in terms of a Riemannian metric and a 1-form. We show that Einstein square metrics can be classified up to the classification of Einstein Riemannian metrics.

Defines a new Randers metric based on an existing one.

Study Kähler-Einstein metrics on singular varieties, proving metric completion properties.

Investigates O(n)-invariant metrics on SPD matrices, extending kernel metrics.

The current paper deals with some new classes of Finsler metrics with reversible geodesics. We construct weighted quasi-metrics associated with these metrics. Further, we investigate some important geometric properties of weighted quasi-metric space. Finally, we discuss the embedding of quasi-metric spaces with general…

Douglas metrics are metrics with vanishing Douglas curvature which is an important projective invariant in Finsler geometry. To find more Douglas metrics, in this paper we consider a class of Finsler metrics called general $(α,β)$-metrics, which are defined by a Riemannian metric $α=\sqrt{a_{ij}(x)y^iy^j}$ and a $1$-fo…

Proposes a method to select fair performance metrics through metric elicitation.

In this paper, we introduce the notion of Einstein-reversibility for Finsler met- rics. We study a class of p-power Finsler metrics determined by a Riemann metric and 1-form which are of Einstein-reversibility. It shows that such a class of Finsler metrics of Einstein-reversibility are always Einstein metrics. In parti…

In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in $\boldsymbol{R}^3$ are Kossowski metrics, and t…

Study rough Riemannian metrics on manifolds, proving their connectedness and completeness.

Introduces new metric for Riemannian metrics, extending unbalanced optimal transport.

Generalizes Thurston's asymmetric metric to flat metrics.

In this paper, we study generalized Douglas-Weyl $(α, β)$-metrics. Suppose that an regular $(α, β)$-metric $F$ is not of Randers type. We prove that $F$ is a generalized Douglas-Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover by ignoring the regularity, if $F$ is not a Berwald met…

The paper introduces two new metrics on outer space and shows fixed points for their actions.

This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the $L^2$ Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold. The main body of the thesis is a description of the completion manifold of metrics with respect to the $L^…

The paper examines Finsler metrics under Ricci flow and finds they are Einstein.

In this note, we consider two Riemannian metrics on a moduli space of metric graphs. Each of them could be thought of as an analogue of the Weil-Petersson metric on the moduli space of metric graphs. We discuss and compare geometric features of these two metrics with the "classic" Weil-Petersson metric in Teichmüller t…

Paper shows regions close to negatively curved metrics are minimal fillings and rigid.

Given any two Einstein (pseudo-)metrics, with scalar curvatures suitably related, we give an explicit construction of a Poincaré-Einstein (pseudo-)metric with conformal infinity the conformal class of the product of the initial metrics. We show that these metrics are equivalent to ambient metrics for the given conforma…

In this paper, we study a class of Finsler metrics which contains the class of P-reducible metrics. Finsler metrics in this class are called generalized P-reducible metrics. We consider generalized P-reducible metrics with scalar flag curvature and find a condition under which these metrics reduce to C-reducible metric…

The paper defines and analyzes Kähler metrics near a compact manifold, showing their deviation from Poincaré-type metrics.

Characterizes metrics on Lie groups, proving non-simultaneous existence of balanced and pluriclosed metrics.

Paper defines Fisher co-metric on cotangent bundle and clarifies its relation to variance.

The paper studies conformally flat cubic metrics with isotropic curvature, finding they must be Minkowski.

We study the Lipschitz metric on Teichmuller space (defined by Thurston) and compare it with the Teichmuller metric. We show that in the thin part of Teichmuller space the Lipschitz metric is approximated up to bounded additive distortion by the sup metric on a product of lower-dimensional spaces (similar to the Teichm…

Study on a metric on Hermitian metrics space, proving diffeomorphisms and completeness.

Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle …