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The Harmonic Variational Principle for the Einstein-Hilbert Functional
Sergey 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 metri…
An Inequality Comparing the Dirichlet Energy and the Bienergy of Maps Between Riemannian Manifolds
Sergey Stepanov, Irina Tsyganok
We establish a geometric inequality relating the Dirichlet energy and the bienergy of smooth maps \[ f : (M,g) \to (\overline{M},\overline{g}) \] between Riemanni…
Localized Curvature Domination and Rigidity of Harmonic Maps
Sergey Stepanov
We establish a localized Bochner-type rigidity theorem for harmonic maps between Riemannian manifolds. Let be a harmonic map from a comp…
Extensions and Applications of Stein-Weiss Operators to the Study of Traceless Symmetric Tensors
Sergey Stepanov, Irina Tsyganok
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…
Back to harmonic mappings of compact Riemannian manifolds
Sergey Stepanov, Irina Tsyganok
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 t…
Geometric Interpretations and Applications of the Berger-Ebin and York -Orthogonal Decompositions
Sergey Stepanov, Irina Tsyganok
The Berger-Ebin and York -orthogonal decompositions of the vector space of symmetric bilinear differential two-forms are fundamental tools in global Riemannian geometry. In th…
Some numerical characteristic inequalities of compact Riemannian manifolds
Sergey Stepanov, Irina Tsyganok
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…
On the theory of minimal submanifolds and harmonic maps from the point of view of the generalized Bochner technique
Sergey Stepanov, Irina Tsyganok
In the present paper, we study harmonic mappings of complete Riemannian manifolds, as well as minimal and stable minimal submanifolds of complete Riemannian manifolds. We examine c…
On minimal hypersurfaces in Euclidean spaces and Riemannian manifolds
Josef Mikes, Sergey Stepanov, Irina Tsyganok
This paper establishes the conditions under which minimal and stable minimal hypersurfaces are characterized as hyperplanes in Euclidean spaces and as totally geodesic submanifolds…
On a Laplacian which acts on symmetric tensors
J. Mikesh, S. E. Stepanov, I. I. Tsyganok
In the present paper we show properties of a little-known Laplacian operator acting on symmetric tensors. This operator is an analogue of the well known Hodge-de Rham Laplacian whi…