https://neckpinch.com/authors/nurlan-abiev/New arXiv papers by Nurlan Abiev, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2508.07391arxiv:2508.07391Sun, 10 Aug 2025 00:00:00 GMTNurlan A. Abiev — This article deals with the problems of preserving the Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow (NRF). We found out infinitely many generalized Wallach spaces (GWS) on which the positivity of the Ricci curvature of metrics is preserved when evolved by the NRF. Analogously, the number of GWS is infinite as well, when the positivity of the Ricci curvature can be lost. We also obtain some refinements to our previous results devoted to the case of coincided parameters. A series of examples is discussed.Ricci flowHomogeneous spaces https://arxiv.org/abs/2412.03170arxiv:2412.03170Wed, 04 Dec 2024 00:00:00 GMTNurlan Abiev — We proved that on every Stiefel manifold V₂ℝⁿ≅ SO(n)/SO(n-2) with n≥ 3 the normalized Ricci flow preserves the positivity of the Ricci curvature of invariant Riemannian metrics with positive Ricci curvature. Moreover, the normalized Ricci flow evolves all metrics with mixed Ricci curvature into metrics with positive Ricci curvature in finite time. From the point of view of the theory of dynamical systems we proved that for every invariant set Σ of the normalized Ricci flow on V₂ℝⁿ defined as x₁ⁿ⁻²x₂ⁿ⁻²x₃=c, c>0, there exists a smaller invariant set Σ∩ 𝓡₊ for every n≥ 3, where 𝓡₊ is the domain in ℝ₊³ responsible for parameters x₁, x₂, x₃>0 of invariant Riemannian metrics on V₂ℝⁿ admitting positive Ricci curvature.Ricci flowSingularities https://arxiv.org/abs/2409.02570arxiv:2409.02570Wed, 04 Sep 2024 00:00:00 GMTNurlan Abiev — We proved that the normalized Ricci flow does not preserve the positivity of Ricci curvature of Riemannian metrics on every generalized Wallach space with a₁+a₂+a₃≤ 1/2, in particular on the spaces SU(k+l+m)/SU(k)× SU(l) × SU(m) and Sp(k+l+m)/Sp(k)× Sp(l) × Sp(m) independently on k,l and m. The positivity of Ricci curvature is preserved for all original metrics with Ric>0 on generalized Wallach spaces a₁+a₂+a₃> 1/2 if the conditions 4(aⱼ+aₖ)²≥ (1-2aᵢ)(1+2aᵢ)⁻¹ hold for all i,j,k\=\1,2,3\. We also established that the spaces SO(k+l+m)/SO(k)× SO(l)× SO(m) satisfy the above conditions for maxk,l,m\≤ 11, moreover, additional conditions were found to keep Ric>0 in cases when maxk,l,m\≤ 11 is violated. Similar questions have also been studied for all other generalized Wallach spaces given in the classification of Yuri\uı Nikonorov.Ricci flow https://arxiv.org/abs/2402.11692arxiv:2402.11692Sun, 18 Feb 2024 00:00:00 GMTNurlan Abiev — Sets related to positively curved invariant Riemannian metrics on generalized Wallach spaces are considered. The problem arises in studying of the evolution of such metrics under the normalized Ricci flow equation. For Riemannian metrics of the Wallach spaces SU(3)/Tₘₐₓ, Sp(3)/ (Sp(1))³ and F₄/Spin(8) which admit positive sectional curvature and belong to a given invariant surface Σ of the normalized Ricci flow we established that they form a set bounded by three connected and pairwise disjoint regular space curves such that each of them approaches two others asymptotically at infinity. Analogously, for all generalized Wallach spaces the set of Riemannian metrics which belong to Σ and admit positive Ricci curvature is bounded by three curves each consisting of two connected components as regular curves. Intersections and asymptotical behaviors of these components were studied as well.Ricci flowSphere theorems https://arxiv.org/abs/2312.09706arxiv:2312.09706Fri, 15 Dec 2023 00:00:00 GMTNurlan Abiev — We study the behavior of a three-dimensional dynamical system with respect to some set S given in 3-dimensional euclidian space. Geometrically such a system arises from the normalized Ricci flow on some class of generalized Wallach spaces that can be described by a real parameter a∈ (0,1/2), as for S it represents the set of invariant Riemannian metrics of positive sectional curvature on the Wallach spaces. Establishing that S is bounded by three conic surfaces and regarding the normalized Ricci flow as an abstract dynamical system we find out the character of interrelations between that system and S for all a∈ (0,1/2). These results can cover some well-known results, in particular, they can imply that the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature on the Wallach spaces corresponding to the cases a∈ \1/9, 1/8, 1/6\ of generalized Wallach spaces.Ricci flow3-manifoldsSphere theorems