arXiv subjects

Mikhail Karpukhin

Publications and source records attributed to Mikhail Karpukhin.

30 records · Page 2Linked to original sources

arXiv2020

Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the $k$-th perimeter-normalised Steklov eigenvalue is $8πk$, which is the best upper bound for the $k$-th area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realising a weighted Neumann problem as a limit of Steklov problems on perforated domains. For $k = 1$, the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.

math.SP↗

arXiv2020

On the Friedlander-Nadirashvili invariants of surfaces

Let $M$ be a closed smooth manifold. In 1999, L. Friedlander and N. Nadirashvili introduced a new differential invariant $I_1(M)$ using the first normalized nonzero eigenvalue of the Lalpace-Beltrami operator $Δ_g$ of a Riemannian metric $g$. They defined it taking the supremum of this quantity over all Riemannian metrics in each conformal class, and then taking the infimum over all conformal classes. By analogy we use $k$-th eigenvalues of $Δ_g$ to define the invariants $I_k(M)$ indexed by positive integers $k$. In the present paper the values of these invariants on surfaces are investigated. We show that $I_k(M)=I_k(\mathbb{S}^2)$ unless $M$ is a non-orientable surface of even genus. For orientable surfaces and $k=1$ this was earlier shown by R. Petrides. In fact L. Friedlander and N. Nadirashvili suggested that $I_1(M)=I_1(\mathbb{S}^2)$ for any surface $M$ different from $\mathbb{RP}^2$. We show that, surprisingly enough, this is not true for non-orientable surfaces of even genus, for such surfaces one has $I_k(M)>I_k(\mathbb{S}^2)$. We also discuss the connection between the Friedlander-Nadirashvili invariants and the theory of cobordisms, and conjecture that $I_k(M)$ is a cobordism invariant.

math.DG↗

arXiv2020

Conformally maximal metrics for Laplace eigenvalues on surfaces

The paper is concerned with the maximization of Laplace eigenvalues on surfaces of given volume with a Riemannian metric in a fixed conformal class. A significant progress on this problem has been recently achieved by Nadirashvili-Sire and Petrides using related, though different methods. In particular, it was shown that for a given $k$, the maximum of the $k$-th Laplace eigenvalue in a conformal class on a surface is either attained on a metric which is smooth except possibly at a finite number of conical singularities, or it is attained in the limit while a "bubble tree" is formed on a surface. Geometrically, the bubble tree appearing in this setting can be viewed as a union of touching identical round spheres. We present another proof of this statement, developing the approach proposed by the second author and Y. Sire. As a side result, we provide explicit upper bounds on the topological spectrum of surfaces.

math.DG↗

arXiv2019

On the Yang-Yau inequality for the first Laplace eigenvalue

In a seminal paper published in 1980, P. C. Yang and S.-T. Yau proved an inequality bounding the first eigenvalue of the Laplacian on an orientable Riemannian surface in terms of its genus $γ$ and the area. The equality in Yang-Yau's estimate is attained for $γ=0$ by an old result of J. Hersch and it was recently shown by S. Nayatani and T. Shoda that it is also attained for $γ=2$. In the present article we combine techniques from algebraic geometry and minimal surface theory to show that Yang-Yau's inequality is strict for all genera $γ> 2$. Previously this was only known for $γ=1$. In the second part of the paper we apply Chern-Wolfson's notion of harmonic sequence to obtain an upper bound on the total branching order of harmonic maps from surfaces to spheres. Applications of these results to extremal metrics for eigenvalues are discussed.

math.DG↗

arXiv2019

Index of minimal spheres and isoperimetric eigenvalue inequalities

In the present paper we use twistor theory in order to solve two problems related to harmonic maps from surfaces to Euclidean spheres $\mathbb{S}^n$. First, we propose a new approach to isoperimetric inequalities based on energy index. Using this approach we show that for any positive $k$, the $k$-th non-zero eigenvalue of the Laplacian on the real projective plane endowed with a metric of unit area, is maximized on the sequence of metrics converging to a union of $(k-1)$ identical copies of round sphere and a single round projective plane. This extends the results of P. Li and S.-T. Yau for $k=1$ (1982); N. Nadirashvili and A. Penskoi for $k=2$ (2018); and confirms the conjecture made in [KNPP]. Second, we improve the known upper bounds for the area index of minimal two-dimensional spheres and minimal projective planes in $\mathbb{S}^n$. In the course of the proof we establish a twistor correspondence for Jacobi fields, which could be of independent interest for the study of moduli space of harmonic maps.

math.DG↗

arXiv2019

Conformal invariants from nodal sets II. Manifolds with boundary

In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators on manifolds with boundary. We also consider applications to curvature prescription problems on manifolds with boundary. We relate Dirichlet and Neumann eigenvalues and put the results developed here for the Escobar problem into the more general framework of boundary operators of arbitrary order.

math.AP↗

arXiv2019

An isoperimetric inequality for Laplace eigenvalues on the sphere

We show that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, is maximized in the limit by a sequence of metrics converging to a union of k touching identical round spheres. This proves a conjecture posed by the second author in 2002 and yields a sharp isoperimetric inequality for all nonzero eigenvalues of the Laplacian on a sphere. Earlier, the result was known only for k=1 (J. Hersch, 1970), k=2 (N. Nadirashvili, 2002; R. Petrides, 2014) and k=3 (N. Nadirashvili and Y. Sire, 2017). In particular, we argue that for any k>=2, the supremum of the k-th nonzero eigenvalue on a sphere of unit area is not attained in the class of Riemannian metrics which are smooth outsitde a finite set of conical singularities. The proof uses certain properties of harmonic maps between spheres, the key new ingredient being a bound on the harmonic degree of a harmonic map into a sphere obtained by N. Ejiri.

math.DG↗

arXiv2018

On branched minimal immersions of surfaces by first eigenfunctions

It was proved by Montiel and Ros that for each conformal structure on a compact surface there is at most one metric which admits a minimal immersion into some unit sphere by first eigenfunctions. We generalize this theorem to the setting of metrics with conical singularities induced from branched minimal immersions by first eigenfunctions into spheres. Our primary motivation is the fact that metrics realizing maxima of the first non-zero Laplace eigenvalue are induced by minimal branched immersions into spheres. In particular, we show that the properties of such metrics induced from $\mathbb{S}^2$ differ significantly from the properties of those induced from $\mathbb{S}^m$ with $m>2$. This feature appears to be novel and needs to be taken into account in the existing proofs of the sharp upper bounds for the first non-zero eigenvalue of the Laplacian on the $2$-torus and the Klein bottle. In the present paper we address this issue and give a detailed overview of the complete proofs of these upper bounds following the works of Nadirashvili, Jakobson-Nadirashvili-Polterovich, El Soufi-Giacomini-Jazar, Nadirashvili-Sire and Petrides.

math.SP↗

arXiv2018

Maximal metrics for the first Steklov eigenvalue on surfaces

In recent years, eigenvalue optimization problems have received a lot of attention, in particular, due to their connection with the theory of minimal surfaces. In the present paper we prove that on any orientable surface there exists a smooth metric maximizing the first normalized Steklov eigenvalue. For surfaces of genus zero, this has been earlier proved by A. Fraser and R. Schoen. Our approach builds upon their ideas and further developments due to R. Petrides. As a corollary, we show that there exist free boundary branched minimal immersions of an arbitrary compact orientable surface with boundary into a Euclidean ball of some dimension.

math.DG↗

arXiv2017

Steklov problem on differential forms

In this paper we study spectral properties of Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $Λ$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there.We investigate properies of eigenvalues of $Λ$ and prove a Hersch-Payne-Schiffer type inequality relating products of those eigenvalues to eigenvalues of Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of $Λ$ are always at least as large as eigenvalues of Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of $p$-forms on the boundary of $2p+2$-dimensional manifold shares a lot of important properties with the classical Steklov eigenvalue problem on surfaces.

math.SP↗

arXiv2013

Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

We prove two explicit bounds for the multiplicities of Steklov eigenvalues $σ_k$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given smooth Riemannian surface with boundary, the multiplicities of Steklov eigenvalues $σ_k$ are uniformly bounded in $k$.

math.DG↗

arXiv2013

Spectral properties of a family of minimal tori of revolution in five-dimensional sphere

The normalized eigenvalues $Λ_i(M,g)$ of the Laplace-Beltrami operator can be considered as functionals on the space of all Riemannian metrics $g$ on a fixed surface $M$. In recent papers several explicit examples of extremal metrics were provided. These metrics are induced by minimal immersions of surfaces in $\mathbb{S}^3$ or $\mathbb{S}^4$. In the present paper a family of extremal metrics induced by minimal immersions in $\mathbb{S}^5$ is provided.

math.DG↗