arXiv research

Spherically symmetric metrics form a rich and important class of metrics. Many well-known Finsler metrics of constant flag curvature can be locally expressed as a spherically symmetric metric on R^n. In this paper, we study spherically symmetric metrics with constant Ricci curvature and constant flag curvature.

A Riemannian metric is of constant curvature if and only if it is locally projectively flat. There are infinitely many locally projectively flat Finsler metrics of constant curvature, that are special solutions to the Hilbert's Fourth Problem. In this paper, we use the technique in the paper titled "Finsler metrics wit…

The study explores metrics with constant curvature on compact manifolds.

Constructs Kähler metrics with constant scalar curvature on blown-up manifolds.

Solves geodesic equations on specific metrics types.

Complete constant positive scalar curvature metrics on S^n - {p_1, ..., p_k} admit a definite asymptotic structure; i.e. the metric is asymptotic to a specific S^{n-1}-invariant metric near the puncture points. This allows one to glue together two such metrics near their puncture points, provided the asymptotic structu…

Study constant Q-curvature metrics on conic 4-manifolds.

Compact quasi-Einstein metrics with constant scalar curvature are locally homogeneous in 3D.

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

Study spherically symmetric Finsler metrics with specific curvature properties.

Constructs metrics with negative constant scalar curvature.

Paper builds singular metrics with constant Q-curvature.

Paper studies metrics with constant Q-curvature near singular points.

We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe…

Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.

Constructs metrics on Riemann surfaces with singularities.

Classifies Kähler metrics with constant holomorphic curvature.

This paper is concerned with the existence of constant scalar curvature Kaehler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kaehler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which already car…

New Finsler metrics with constant flag curvature defined using Weyl-type curvature tensor.

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $n\geq 3$. We prove the existence of such conformal metrics in the cases of $n=6,7$ or the manifold is spin and some other remai…

We establish several nonuniqueness results for the problem of finding complete conformal metrics with constant (fourth-order) $Q$-curvature on compact and noncompact manifolds of dimension $\geq5$. Infinitely many branches of metrics with constant $Q$-curvature, but without constant scalar curvature, are found to bifur…

The local structure of Finsler metrics of constant flag curvature have been historically mysterious. It is proved that every Matsumoto metric of constant flag curvature on a manifold of dimension n \geq 3 is either Riemannian or locally Minkowskian.

Study on Hermitian metrics and curvature properties of complex manifolds.

New metrics found without topological restrictions.

Study on constant mu-scalar curvature Kähler metrics, generalizing cscK and Kähler-Ricci solitons.

Ruled surfaces with Ricci metrics use curves of constant torsion.

Compact metrics found with specific curvature properties on 3D surfaces.

Classifies metrics with specific curvature properties on a ball.

In [7], a notion of constant scalar curvature metrics on piecewise flat manifolds is defined. Such metrics are candidates for canonical metrics on discrete manifolds. In this paper, we define a class of vertex transitive metrics on certain triangulations of $\mathbb{S}^3$; namely, the boundary complexes of cyclic polyt…

Extends Lipschitz functions while preserving local constants.

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

Unique conformal metrics found on certain manifolds.

A comparison theorem for the isoperimetric profile on the universal cover of surfaces evolving by normalised Ricci flow is proven. For any initial metric, a model comparison is constructed that initially lies below the profile of the initial metric and which converges to the profile of the constant curvature metric. Th…

We investigate projective spherically symmetric Finsler metrics with constant flag curvature in $R^n$ and give the complete classification theorems. Furthermore, a new class of Finsler metrics with two parameters on n-dimensional disk are found to have constant negative flag curvature.

In Finsler geometry, there are infinitely many models of constant curvature. The Funk metrics, the Hilbert-Klein metrics and the Bryant metrics are projectively flat with non-zero constant curvature. A recent example constructed by the author is projectively flat with zero curvature. In this paper, we introduce a techn…

Establishes geodesic stability for Kähler metrics, proving existence of constant scalar curvature.

Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we classify a class of singular $(α,β)$-metrics which are locally projectively flat with constant flag curvature in dimension $n= 2$ and $n \ge 3$ respectively. Further, we determine t…

The paper finds conditions for Yamabe solitons' metrics to be constant scalar curvature.

Solves geodesic equations on specific metrics.

Constructed static vacuum metrics in 5D with negative cosmological constant.

New metrics found on orbifold resolutions with specific curvature.

The nonlinear equations describing all the nonsingular pencils of metrics of constant Riemannian curvature are derived and the integrability of these nonlinear equations by the method of inverse scattering problem is proved. It is proved that all the nonsingular pairs of compatible metrics of constant Riemannian curvat…

Concrete example of 2-sphere with constant curvature 1 and closed geodesics.

This paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature/

Estimates for metrics with constant Chern scalar curvature on complex manifolds.

New metric measure space theory for Lipschitz constants.

We study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We …

Let $(M,\,g)$ be a Poincar$\acute{\text{e}}$-Einstein manifold with a smooth defining function. In this note, we prove that there are infinitely many asymptotically hyperbolic metrics with constant $Q$-curvature in the conformal class of an asymptotically hyperbolic metric close enough to $g$. These metrics are paramet…