https://neckpinch.com/authors/sergey-stepanov/New arXiv papers by Sergey Stepanov, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2608.08233arxiv:2608.08233Sat, 08 Aug 2026 00:00:00 GMTSergey Stepanov, Irina Tsyganok — Let (M,g) be a compact n-dimensional Riemannian manifold, n>2. We introduce a restricted variational principle for the Einstein-Hilbert functional by requiring the admissible metric variations to satisfy the harmonic gauge condition. We derive the corresponding Euler-Lagrange equation and show that a metric is critical with respect to all volume-preserving harmonic variations if and only if its Einstein tensor differs from a multiple of the metric by an element of the image of the adjoint Bianchi operator. We prove that every harmonic critical metric determines a compact Ricci soliton whose soliton constant is given by the normalized Einstein-Hilbert functional. By Perelman's theorem, every such metric is in fact the metric of a compact gradient Ricci soliton. Conversely, every compact gradient Ricci soliton satisfies the restricted Euler-Lagrange equation. Thus, a compact Riemannian metric is harmonic critical if and only if it is the metric of a compact gradient Ricci soliton. We further show that the gauge one-form differs from the negative differential of a soliton potential by a Killing one-form. In particular, if the Ricci tensor is negative definite, then the gauge one-form vanishes and the metric is Einstein. Moreover, every non-Einstein harmonic critical metric is necessarily shrinking.Ricci solitons https://arxiv.org/abs/2605.25723arxiv:2605.25723Mon, 25 May 2026 00:00:00 GMTSergey Stepanov — The stability and deformation theory of Einstein metrics traditionally relies on the classical Berger-Ebin transverse-traceless gauge, which structurally decouples the scalar trace from the divergence-free component of metric perturbations. In the present paper, we introduce a new spectral-geometric framework based on the Chen-Nagano gauge condition. This condition naturally arises from the harmonicity of the identity map and is intrinsically satisfied by the Ricci tensor itself via the contracted second Bianchi identity. Unlike the classical transverse-traceless framework, the Chen-Nagano gauge preserves a nontrivial interaction between the trace and trace-free sectors of a deformation. We establish a first-order differential relation proving that the divergence of the trace-free part is completely governed by the gradient of the scalar trace. Utilizing commutation formulas on Einstein manifolds, we derive a second-order spectral coupling relation that links the Lichnerowicz Laplacian to a shifted scalar operator. As a primary geometric consequence, we prove that under suitable spectral pinching assumptions, the Chen-Nagano gauge collapses to the classical transverse-traceless gauge. Specifically, we show that on compact connected negatively curved Einstein manifolds, any volume-preserving Chen-Nagano harmonic deformation whose trace-free component lies below a specific spectral threshold determined by the Einstein constant is necessarily transverse-traceless. Furthermore, we connect this rigidity to the curvature operator of the second kind, establishing explicit lower spectral bounds. Finally, we provide a dynamical interpretation within the Ricci flow framework, demonstrating that the linearized Ricci flow under the Chen-Nagano gauge reduces to a strictly parabolic equation governed by the Lichnerowicz Laplacian, ensuring exponential decay of admissible perturbations.Ricci flowEinstein metricsCurvature pinchingStability & uniquenessEigenvalues & spectrum https://arxiv.org/abs/2304.03584arxiv:2304.03584Fri, 07 Apr 2023 00:00:00 GMTVladimir Rovenski, Sergey Stepanov, Irina Tsyganok — In the paper, we study complete almost Ricci solitons using the concepts and methods of geometric dynamics and geometric analysis. In particular, we characterize Einstein manifolds in the class of complete almost Ricci solitons. Then, we examine compact almost Ricci solitons using the orthogonal expansion of the Ricci tensor, this allows us to substantiate the concept of almost Ricci solitons.Ricci solitonsEinstein metricsAlmost & η-Ricci solitons https://arxiv.org/abs/2302.02799arxiv:2302.02799Mon, 06 Feb 2023 00:00:00 GMTSergey E. Stepanov, Josef Mikes, Irina I. Tsyganok — In this article, we consider an orthogonal decomposition of the traceless part of the Ricci tensor of a compact Riemannian manifold and study its application to the geometry of compact almost Ricci solitons. In addition, we consider an orthogonal expansion of the traceless part of the second fundamental form of a compact spacelike hypersurface in a Lorentzian manifold and study its application to the problem of constructing solutions of general relativistic constraint equations in vacuum. In these two cases, we use the well-known Ahlfors Laplacian.Ricci solitonsAlmost & η-Ricci solitonsPhysics