arXiv research

New Einstein metric found on non-standard solvmanifold.

The paper studies geodesic orbit properties in Finsler spaces.

In this paper, we introduce the notion of standard homogeneous $(α_1,α_2)$-metrics, as a natural non-Riemannian deformation for the normal homogeneous Riemannian metrics. We prove that with respect to the given bi-invariant inner product and orthogonal decompositions for $\mathfrak{g}$, if there exists one generic stan…

Study minimal networks on spheres and balls near standard metrics.

The paper classifies compact homogeneous Finsler manifolds with positive flag curvature.

Optimizes sharp curvature inequality on spheres, proving near-minimizers are close to standard metric.

Computed p-widths for real projective plane.

In this review, we collect several results for conformally standard stationary spacetimes (SxR,g) obtained in terms of a Finsler metric of Randers type on the orbit manifold S that we call Fermat metric. This metric is obtained by applying the relativistic Fermat principle and it turns out that it encodes all the causa…

In this paper we relate the Fefferman-Graham ambient metric construction for conformal manifolds to the approach to conformal geometry via the canonical Cartan connection. We show that from any ambient metric that satisfies a weakening of the usual normalisation condition, one can construct the conformal standard tract…

Study on Ricci flow on 4-spheres, proving standard sphere convergence.

New theorem on spheres with punctures using infinity metric.

Unified framework for comparing classification metrics across different imbalance rates.

The paper constructs Einstein Sasaki metrics on solvable Lie groups.

We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standar…

The paper studies stability of Einstein metrics on non-simple Lie group homogeneous spaces.

Optimal controls for conformal Laplacian obstacle problems on spheres and manifolds.

We study the systolic area (defined as the ratio of the area over the square of the systole) of the 2-sphere endowed with a smooth riemannian metric as a function of this metric. This function, bounded from below by a positive constant over the space of metrics, have the standard metric $g\_0$ for critic point, althoug…

Theorem proves spectral rigidity of warped product metrics.

Characterizes invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.

Let $\mathcal{C}$ be the space of smooth metrics $g$ on a given compact manifold $M^{n}$ ($n\geq3$) with constant scalar curvature and unitary volume. The goal of this paper is to study the critical point of the total scalar curvature functional restricted to the space $\mathcal{C}$ (we shall refer to this critical poi…

We show that a metric of arbitrary dimension and signature which allows for a standard Wick-rotation to a Riemannian metric necessarily has a purely electric Riemann and Weyl tensor.

Comparisons on $L^{n\over 2}$-norms of scalar curvatures between Riemannian metrics and standard metrics are obtained. The metrics are restricted to conformal classes or under certain curvature conditions.

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In…

It is shown that parts of planes, helicoids and hyperbolic paraboloids are the only minimal surfaces ruled by geodesics in the three dimensional Riemannian Heisenberg group. It is also shown that they are the only surfaces in the three dimensional Heisenberg group whose mean curvature is zero with respect to both of th…

We study the space of Riemannian metrics with positive scalar curvature on a compact manifold with boundary. These metrics extend a fixed boundary metric and take a product structure on a collar neighbourhood of the boundary. We show that the weak homotopy type of this space is preserved by certain surgeries on the bou…

We study solutions to the static vacuum Einstein equations on exterior domains with prescribed metric and mean curvature on the inner boundary. It is proved that for any such boundary data near the standard round boundary data in Euclidean space, there exists a unique AF solution to the static vacuum equations realizin…

In the paper "Einstein metrics on compact simple Lie groups attached to standard triples", the authors introduced the definition of standard triples and proved that every compact simple Lie group $G$ attached to a standard triple $(G,K,H)$ admits a left-invariant Einstein metric which is not naturally reductive except …

We construct pairs of conformally equivalent isospectral Riemannian metrics $φ_1 g$ and $φ_2 g$ on spheres $S^n$ and balls $B^{n+1}$ for certain dimensions $n$, the smallest of which is $n=7$, and on certain compact simple Lie groups. In the case of Lie groups, the metric $g$ is left-invariant. In the case of spheres a…

Study the stability of Einstein metrics on homogeneous spaces.

This research proposes a new (old) metric for evaluating goodness of fit in topic models, the coefficient of determination, or $R^2$. Within the context of topic modeling, $R^2$ has the same interpretation that it does when used in a broader class of statistical models. Reporting $R^2$ with topic models addresses two c…

Short note finds a new metric from sphere quotients.

A Riemannian Einstein solvmanifold is called standard, if the orthogonal complement to the nilradical of its Lie algebra is abelian. No examples of nonstandard solvmanifolds are known. We show that the standardness of an Einstein metric solvable Lie algebra is completely detected by its nilradical and prove that many c…

Despite the growing importance of multilingual aspect of web search, no appropriate offline metrics to evaluate its quality are proposed so far. At the same time, personal language preferences can be regarded as intents of a query. This approach translates the multilingual search problem into a particular task of searc…

Evaluation metrics for prediction models don't fully reflect intervention impact.

We show that many standard results of Lorentzian causality theory remain valid if the regularity of the metric is reduced to $C^{1,1}$. Our approach is based on regularisations of the metric adapted to the causal structure.

Neural network pruning lacks standardized benchmarks and metrics.

Study critical metrics on manifolds, proving specific isometries.

Unified theory for adaptive image convolutions using metric perspectives.

It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and particularly to study length-minimizing curves where neither classical techniques such as…

In this note I study the Sasakian geometry associated to the standard CR structure on the Heisenberg group, and prove that the Sasaki cone coincides with the set of extremal Sasakian structures. Moreover, the scalar curvature of these extremal metrics is constant if and only if the metric has $Φ$-sectional curvature $-…

Unique conformal metrics found on certain manifolds.

The study finds counterexamples to curvature estimates for minimizing surfaces.

Study Einstein metrics on $H imes H/ΔK$ spaces.

Study finds infinite families of Sasaki-Einstein metrics on spheres.

In this article we investigate deformations of a scalar-flat Kähler metric on the total space of complex line bundles over CP^1 constructed by C. LeBrun. In particular, we find that the metric is included in a one-dimensional family of such metrics on the four-manifold, where the complex structure in the deformation is…

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime is equal to the index of its spatial projection as a geodesic of a Finsler metric associated to the spacetime. Moreover we obtain the Morse relations of lightlike geodesics connecting a point to an integral line of the standar…

We construct isospectral pairs of Riemannian metrics on S^5 and on B^6, thus lowering by three the dimension of spheres and balls on which such metrics have been constructed previously (S^{n\ge 8} and B^{n\ge 9}). We also construct continuous families of isospectral Riemannian metrics on S^7 and on B^8. In each of thes…

We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption …