arXiv research

We define a partition of the space of projectively flat metrics in three classes according to the sign of the Chern scalar curvature; we prove that the class of negative projectively flat metrics is empty, and that the class of positive projectively flat metrics consists precisely of locally conformally flat-Kähler met…

Study of flat metrics on orbifolds and their moduli spaces.

Investigate AR-Finsler metrics for local dual flatness and projective flatness.

We propose two conjectures about Ricci-flat metrics: Conjecture 1: A Ricci-flat projectively induced metric is flat. Conjecture 2: A Ricci-flat metric on an $n$-dimensional complex manifold such that the $a_{n+1}$ coefficient of the TYZ expansion vanishes is flat. We verify Conjecture 1 (see Theorem 1.1) under the assu…

Study describes flat metric moduli spaces on 4D manifolds.

In this paper we consider flat metrics (semi-translation structures) on surfaces of finite type. There are two main results. The first is a complete description of when a set of simple closed curves is spectrally rigid, that is, when the length vector determines a metric among the class of flat metrics. Secondly, we gi…

Study cylindrical symmetric Finsler metrics that are projectively flat.

In this work, the dual flatness, which is connected with Statistics and Information geometry, of general $(α,β)$-metrics (a new class of Finsler metrics) is studied. A nice characterization for such metrics to be dually flat under some suitable conditions is provided and all the solutions are completely determined. By …

The well-known Funk metric F(x,y) is projectively flat with constant flag curvature K=-1/4 and the Hilbert metric H(x,y):=(F(x,y)+F(x,-y))/2 is projectively flat with constant curvature K=-1. These metrics are the special solutions to Hilbert's Fourth Problem. In this paper, we construct a non-trivial R-flat spray usin…

Constructs scalar-flat Kähler metrics with varying conical singularities.

Paper proves flat metrics from holomorphic quadratic differentials can be identified by length spectrum.

There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat …

Constructs metrics with negative curvature on specific manifold types.

Conditions for flat 3-manifolds with diagonal metrics are identified.

New metrics found on non-Kähler Calabi-Yau manifolds.

The class of spherically symmetric Finsler metrics is studied and locally dually flat and projectively flat spherically symmetric Finsler metrics is classified.

Generalizes Thurston's asymmetric metric to flat metrics.

In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked length spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity. The novel proof suggests a…

Study shows tori metrics converging to flat under specific conditions.

In a recent paper Donaldson explains how to use an older construction of Joyce to obtain four dimensional local models for scalar-flat Kahler metrics with a 2-torus symmetry. Using this idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kahler metrics. In…

This article classifies closed G2-structures such that the induced metric is conformally flat. It is shown that any closed G2-structure with conformally flat metric is locally equivalent to one of three explicit examples. In particular, it follows from the classification that any closed G2-structure inducing a metric t…

The paper constructs flat metrics on orbifolds and resolutions.

Counterexample disproves conjecture on flat metrics and fiber bundles.

In this paper, we prove that a strongly convex complex Finsler metric $F$ on a domain $D\subset\mathbb{C}^n$ is projectively flat (resp. dually flat) if and only if $F$ comes from a strongly convex complex Minkowski metric.

In this paper, we study a class of Finsler metrics composed by a Riemann metric $α=\sqrt{a_{ij}(x)y^i y^j}$ and a $1$-form $β=b_i(x)y^i$ called general ($α$, $β$)-metrics. We classify those projectively flat when $α$ is projectively flat. By solving the corresponding nonlinear PDEs, the metrics in this class are totall…

New methods find Ricci-flat metrics on specific Lie groups.

The paper describes flat Hessian metrics on surfaces and their potentials.

The paper classifies special types of contact metric manifolds with curvature conditions.

Flat metrics on hyperbolic surfaces embed as polyhedral surfaces in (2+1)-spacetimes.

Study on Gauduchon manifolds finds metrics for projectively flat bundles.

We show that any asymptotically locally Euclidean (ALE) metric which is obstruction-flat or extended obstruction-flat must be ALE of a certain optimal order. Moreover, our proof applies to very general elliptic systems and in any dimension $n \geq 3$. The proof is based on the technique of Cheeger-Tian for Ricci-flat m…

Study on flat metrics on 3D and 4D manifolds, focusing on topology and algebra.

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…

Study on Calabi-Yau metrics and their degenerations.

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…

Study on how incomplete smooth metrics degenerate from asymptotically conical Ricci-flat Kähler metrics.

The paper studies scalar flat Kähler metrics on line bundles and proves their properties.

The paper constructs a new Ricci-flat metric on almost abelian Lie groups.

The Yamabe flow on flat manifolds converges to a scalar flat metric.

Simply connected moduli space of Ricci flat metrics on K3 surfaces.

Study stability and instability of Ricci-flat metrics under generalized Ricci flow.

In this paper the geometry of normal metric contact pair manifolds is studied under the flatness of conformal, concircular and quasi-conformal curvature tensors. It is proved that a conformal flat normal metric contact pair manifold is an Einstein manifold with a negative scalar curvature and has positive sectional cur…

We classify radial scalar flat metrics with constant third coeffcient of its TYZ expansion. As a byproduct of our analysis we provide a characterization of Simanca's scalar flat metric.

In this note, we obtain existence results for complete Ricci-flat Kahler metrics on crepant resolutions of singularities of Calabi-Yau varieties. Furthermore, for certain asymptotically flat Calabi-Yau varieties, we show that the Ricci-flat metric on the resolved manifold has the same asymptotic behavior as the initial…

We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with boundary, in dimension $n\geq3$. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric $g$, there is a conformally equivalent asymptotically flat scal…

Positive mass theorem for non-smooth metrics on flat manifolds with corners.

Length metrics can be closely approximated by conformally flat metrics.

Study K3 surfaces and their metrics, focusing on dynamics.