arXiv subjects

Mikhail A. Karpukhin

Publications and source records attributed to Mikhail A. Karpukhin.

5 recordsLinked to original sources

arXiv2017

Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds

Consider a compact Riemannian manifold with boundary. In this short note we prove that under certain positive curvature assumptions on the manifold and its boundary the Steklov eigenvalues of the manifold are controlled by the Laplace eigenvalues of the boundary. Additionally, in two dimensions we obtain an upper bound for these Steklov eigenvalues in terms of topology of the surface without any curvature restrictions.

math.DG↗

arXiv2015

Upper bounds for the first eigenvalue of the Laplacian on non-orientable surfaces

In 1980 Yang and Yau~\cite{YY} proved the celebrated upper bound for the first eigenvalue on an orientable surface of genus $γ$. Later Li and Yau~\cite{LY} gave a simple proof of this bound by introducing the concept of conformal volume of a Riemannian manifold. In the same paper they proposed an approach for obtaining a similar estimate for non-orientable surfaces. In the present paper we formalize their argument and improve the bounds stated in~\cite{LY}.

math.DG↗

arXiv2014

Maximization of the first nontrivial eigenvalue on the surface of genus two

The first nontrivial eigenvalue of the Laplacian can be considered as a functional on the space of all Riemannian metrics of unit volume on a fixed surface. In this paper we prove that for the surface of genus 2 the supremum of this functional is equal to $16π$. This provides a positive answer to the conjecture by Jakobson, Levitin, Nadirashvili, Nigam and Polterovich.

math.DG↗

arXiv2014

Non-maximality of known extremal metrics on torus and Klein bottle

El Soufi-Ilias' theorem establishes a connection between minimal submanifolds of spheres and extremal metrics for eigenvalues of the Laplace-Beltrami operator. Recently, this connection was used to provide several explicit examples of extremal metrics. We investigate the maximality of these metrics and prove that all of them are not maximal.

math.SP↗

arXiv2012

Spectral properties of bipolar surfaces to Otsuki tori

The $i$-th eigenvalue $λ_i$ of the Laplace-Beltrami operator on a surface can be considered as a functional on the space of all Riemannian metrics of unit volume on this surface. Surprisingly only few examples of extremal metrics for these functionals are known. In the present paper a new countable family of extremal metrics on the torus is provided.

math.DG↗