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Sergey Stepanov

Publications and source records attributed to Sergey Stepanov.

At least 19 recordsLinked to original sources
arXiv2026

The Harmonic Variational Principle for the Einstein-Hilbert Functional

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.

math.DG↗
arXiv2026

A Restricted Chen-Nagano Variational Principle for the Einstein-Hilbert Functional

This paper introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional on compact Riemannian manifolds. Instead of considering arbitrary symmetric variations of the metric, we restrict the variational problem to an infinite-dimensional subspace determined by the Chen-Nagano gauge constraint. We derive the corresponding restricted Euler-Lagrange equations and obtain a novel structural characterization of critical metrics. The resulting criticality condition is expressed by the equation $E_g = B_g^{*}(\theta) + c\,g$ which may be regarded as a restricted counterpart to the classical Einstein equation. Furthermore, we demonstrate that this variational framework naturally leads to generalized Ricci almost soliton structures and, in the gradient case, to gradient Ricci almost solitons. Several global rigidity consequences of this restricted principle are also established.

math.DG↗
arXiv2026

Spectral Properties of the Chen-Nagano Gauge on Einstein Manifolds

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.

math.DG↗
arXiv2026

Localized Curvature Domination and Rigidity of Harmonic Maps

We establish a localized Bochner-type rigidity theorem for harmonic maps between Riemannian manifolds. Let $f : (M,g) \to (\overline{M},\overline{g})$ be a harmonic map from a compact manifold. Instead of assuming a global nonpositivity condition on the sectional curvature of the target, we impose a curvature bound localized to the image $f(M)$, expressed via the maximal sectional curvature encountered along the image. We prove that if the minimal Ricci curvature of the domain dominates this image-dependent curvature bound in a sharp quantitative pinching inequality involving the maximal energy density of $f$, then the map is constant. At the critical threshold, we obtain a homothetic classification: the differential is parallel and the image is totally geodesic. The result replaces global curvature sign assumptions with an image-dependent curvature domination principle and yields a localized analogue of Yano-Ishihara-type rigidity.

math.DG↗
arXiv2026

Projective Maps from the Perspective of Elliptic Differential Operators

This paper develops an analytical approach to the study of the geometry of projective maps using the theory of elliptic differential operators. We construct two elliptic operators of second and fourth order, whose kernels characterize projective diffeomorphisms between Riemannian manifolds and one-parameter groups of projective diffeomorphisms (transformations) of a Riemannian manifold onto itself, respectively. This approach establishes a natural correspondence between analytical and geometric properties, enabling the study of projective diffeomorphisms via operator-theoretic methods. The proposed framework provides a new understanding of projective structures on Riemannian manifolds and extends classical results in differential geometry.

math.DG↗
arXiv2026

An Inequality Comparing the Dirichlet Energy and the Bienergy of Maps Between Riemannian Manifolds

We establish a geometric inequality relating the Dirichlet energy $E_1(f)$ and the bienergy $E_2(f)$ of smooth maps \[ f : (M,g) \to (\overline{M},\overline{g}) \] between Riemannian manifolds. Assume that $(M,g)$ is a compact, connected Riemannian manifold whose Ricci curvature has global minimum $\operatorname{Ric}_{\min}$, and that the target manifold $(\overline{M},\overline{g})$ has non-positive sectional curvature along $f(M)$. We prove that \[ E_2(f) \ge \operatorname{Ric}_{\min}\, E_1(f). \] We further analyze the equality case and obtain rigidity results: equality holds if and only if $f$ is totally geodesic and of constant rank. Applications to maps into Hadamard manifolds are also presented. To the best of our knowledge, this is the first geometric inequality directly relating the Dirichlet energy and the bienergy of smooth maps. This result establishes a direct connection between the Ricci curvature of the domain and higher-order variational energies.

math.DG↗
arXiv2025

Extensions and Applications of Stein-Weiss Operators to the Study of Traceless Symmetric Tensors

First-order differential operators arising from the representation-theoretic decomposition of the covariant derivative play a central role in Riemannian geometry. In this paper, we study Stein-Weiss $O(n)$-gradients acting on covariant symmetric trace-free tensors of arbitrary rank $p \ge 2$. By analyzing the decomposition of $T^*M \otimes S_0^p(M)$ into its $O(n)$-irreducible components, we explicitly describe the corresponding generalized gradients and compute Weitzenbock formulas for their adjoint compositions. These results extend Bouguignon four-dimensional formulas for $p = 2$ and generalize previous work of other authors to higher-rank symmetric tensors. The formulas obtained provide a unified framework for understanding second-order Stein-Weiss operators and yield tools applicable to deformation complexes, curvature estimates, and stability problems in geometric analysis. The article continues the authors' earlier investigations of Stein-Weiss operators on natural tensor bundles.

math.DG↗
arXiv2025

Back to harmonic mappings of compact Riemannian manifolds

In this paper, we address several interconnected problems in the theory of harmonic maps between Riemannian manifolds. First, we present necessary background and establish one of the main results of the paper: a criterion characterizing when a smooth submersion or diffeomorphism between Riemannian manifolds is harmonic. This result provides a useful analytic condition for verifying the harmonicity of geometric mappings. Second, we investigate the L2-orthogonal decomposition of the pullback metric associated with a harmonic map. We analyze the structure of this decomposition and discuss its geometric implications, particularly in the context of the energy density and trace conditions. Finally, we study harmonic symmetric bilinear forms and harmonic Riemannian metrics. Special attention is given to their role in the theory of harmonic identity maps. We derive new results that link these notions and demonstrate how they contribute to the broader understanding of harmonicity in geometric analysis.

math.DG↗
arXiv2025

Geometric Interpretations and Applications of the Berger-Ebin and York $L^2$-Orthogonal Decompositions

The Berger-Ebin and York $L^2$-orthogonal decompositions of the vector space of symmetric bilinear differential two-forms are fundamental tools in global Riemannian geometry. In this paper, we investigate the structure of Ricci tensors on compact Riemannian manifolds, with a particular focus on compact Ricci almost solitons, utilizing both the Berger-Ebin and York $L^2$-orthogonal decompositions. In addition, we explore applications of the York $L^2$-orthogonal decomposition to the theory of submanifolds and to the study of harmonic maps between Riemannian manifolds.

math.DG↗
arXiv2025

Some numerical characteristic inequalities of compact Riemannian manifolds

In this paper, we consider numerical characteristics of the connected compact Riemannian manifold (M, g) such as the supremum and infimum of the scalar curvature s, Ricci curvature Ric and sectional curvature sec, as well as their applications. Below are two examples of proven results. The first statement: If (M, g) be a connected, compact Riemannian manifold of even dimension n > 3 whose Ricci and sectional curvatures satisfy the strict inequality n Inf (sec) > Sup (Ric), then M is diffeomorphic to the Euclidean n-dimensional sphere of some radius r or the real projective n-dimensional space. The second statement: There is no harmonic immersion of an n-dimensional connected, complete Riemannian manifold (M, g) into the Euclidean n-sphere of radius r if there exists inf(Ric) such that Inf (Ric) > n/2r^2.

math.DG↗
arXiv2024

Optimization of Discrete Parameters Using the Adaptive Gradient Method and Directed Evolution

The problem is considered of optimizing discrete parameters in the presence of constraints. We use the stochastic sigmoid with temperature and put forward the new adaptive gradient method CONGA. The search for an optimal solution is carried out by a population of individuals. Each of them varies according to gradients of the 'environment' and is characterized by two temperature parameters with different annealing schedules. Unadapted individuals die, and optimal ones interbreed, the result is directed evolutionary dynamics. The proposed method is illustrated using the well-known combinatorial problem for optimal packing of a backpack (0-1 KP).

math.OC↗
arXiv2023

Back to almost Ricci solitons

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.

math.DG↗
arXiv2022

Geometry in the large on Hadamard manifolds

In this paper, we prove several Liouville-type theorems on the non-existence of Killing-Yano tensors, Killing tensors, and harmonic symmetric tensors on Hadamard manifolds and, in particular, on Riemannian symmetric spaces of non-compact type. These theorems supplement the well-known vanishing theorems for the above tensors, obtained using the Bochner technique for compact Riemannian manifolds. In turn, the proofs of our theorems will use well-known Liouville-type theorems on the non-existence of subharmonic and harmonic functions on complete Riemannian manifolds, which we have partially modified for the case of Hadamard manifolds and, in particular, Riemannian symmetric spaces of noncompact type.

math.DG↗
arXiv2022

On the optimality of full disclosure

A privately-informed sender can commit to any disclosure policy towards a receiver. We show that full disclosure is optimal under a sufficient condition with some desirable properties. First, it speaks directly to the utility functions of the parties, as opposed to the indirect utility function of the sender; this makes it easily interpretable and verifiable. Second, it does not require the sender's payoff to be a function of the posterior mean. Third, it is weaker than the known conditions for some special cases. With this, we show that full disclosure is optimal under modeling assumptions commonly used in principal-agent papers.

econ.TH↗
arXiv2020

On the Bochner technique for singular distributions

In this paper we continue our recent study of a manifold endowed with a singular or regular distribution, determined as the image of the tangent bundle under a smooth endomorphism, and generalize Bochner's technique to the case of a distribution with a statistical type structure. Following the theory of statistical structures on Riemannian manifolds and construction of an almost Lie algebroid on a vector bundle, we define the modified statistical connection and exterior derivative on tensors. Then we introduce the Weitzenbock type curvature operator on tensors and derive the Bochner-Weitzenbock type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian on a distribution.

math.DG↗