arXiv subjects

Vladimir Medvedev

Publications and source records attributed to Vladimir Medvedev.

13 recordsLinked to original sources

arXiv2026

On static manifolds with boundary admitting a nowhere-vanishing static potential

We study complete static manifolds with boundary admitting a nowhere-vanishing static potential. Our main result shows that, under a natural lower bound relating the scalar curvature and the boundary mean curvature, a simple static manifold with boundary must in fact have positive scalar curvature, negative boundary mean curvature, and be compact; we also obtain explicit relations and estimates involving the volume of the manifold and the geometry of its boundary. In the scalar-flat case, we prove global splitting and Ricci-flat rigidity results, including for disconnected boundary, while in the negative scalar curvature case we establish a sharp mean-curvature bound and characterize the equality case by an exponential warped-product structure. The proofs rely essentially on the study of the associated Einstein manifold. In appendix we derive several identities for static manifolds with boundary and discuss the associated Einstein manifold technique in the boundaryless setting.

math.DG↗

arXiv2026

On free boundary minimal submanifolds with boundary on concentric spheres in Euclidean spac

The search for free boundary minimal submanifolds in Euclidean space with boundaries on a collection of concentric spheres naturally extends the classical problem in the unit Euclidean ball. A key feature of this setting is that the coordinate functions of such submanifolds satisfy a Steklov problem with an indefinite weight. This framework allows us to introduce a spectral index, which in turn yields both upper and lower bounds for the Morse index. As a concrete application, we compute the exact Morse index of an $m$-dimensional flat annulus in an $n$-dimensional spherical shell, showing that it equals $n-m$. Moreover, we study in detail free boundary minimal immersions from 2-dimensional annuli into Euclidean space whose boundaries lie on concentric spheres. We show that the images of these free boundary minimal immersions (FBMI) lie in an $m$-dimensional subspace with $2\leqslant m\leqslant 4$, and list the explicit forms of these FBMI. We also demonstrate how to find examples of FBMIs for which the ratio between the radii of the concentric spheres containing the boundaries is arbitrarily large.

math.DG↗

arXiv2026

On the dimension of the space of static potentials on three-manifolds

We investigate the interplay between the dimension of the space of static potentials and the geometric and topological structure of the underlying static three-manifold. A partial classification of boundaryless static manifolds is obtained in terms of this dimension. We also treat the case of static manifolds with boundary. In particular, we prove that if a compact static manifold with boundary admits a static potential whose zero set is disjoint from the boundary, then the space of static potentials is necessarily one-dimensional. These results rely on a careful analysis of the relative positions of the zero sets of linearly independent static potentials - a technique originally introduced by Miao and Tam.

math.DG↗

arXiv2025

Some rigidity results for static three-manifolds with boundary and positive scalar curvature

This paper studies three-dimensional compact static manifolds with boundary and positive scalar curvature. We prove that, under a suitable bound on the Ricci curvature, the orientable quotient of the Nariai static manifold with boundary $Nar_{-1,1}(\mathbb S^2)$ is the only such manifold with connected boundary, provided that the zero-level set of the potential is connected and does not intersect the boundary. We also establish a rigidity theorem for the upper hemisphere with the standard static potential, in the spirit of Cruz and Nunes.

math.DG↗

arXiv2025

Static manifolds with boundary: Their geometry and some uniqueness theorems

Static manifolds with boundary were recently introduced to mathematics. This kind of manifold appears naturally in the prescribed scalar curvature problem on manifolds with boundary when the mean curvature of the boundary is also prescribed. They are also interesting from the point of view of general relativity. For example, the (time-slice of the) photon sphere on the Riemannian Schwarzschild manifold splits it into static manifolds with boundary. In this paper, we prove a number of theorems that relate the topology and geometry of a given static manifold with boundary to some properties of the zero-level set of its potential (such as connectedness and closedness). Also, we characterize the round ball in the Euclidean 3-space with standard potential as the only scalar-flat static manifold with mean-convex boundary whose zero-level set of the potential has Morse index one. This result follows from a general isoperimetric inequality for 3-dimensional static manifolds with boundary, whose zero-level set of the potential has Morse index one. Finally, we prove some uniqueness theorems for the domains bounded by the photon sphere on the Riemannian Schwarzschild manifold.

math.DG↗

arXiv2025

On electrostatic manifolds with boundary

Static manifolds with boundary were recently introduced by Cruz and Vitório in the context of the prescribed scalar curvature problem in a manifold with boundary with prescribed mean curvature. This kind of manifold is also interesting from the point of view of the general theory of relativity. In this article, we introduce electrostatic manifolds with boundary as a natural generalization of static manifolds with boundary in the presence of a non-zero electric field. We study the geometry of the zero-level set of the potential and its connection to the global properties of electrostatic manifolds with boundary. In particular, we establish some rigidity theorems for the 3-dimensional Euclidean ball and for the Reissner-Nordström manifold.

math.DG↗

arXiv2025

Some new functionals related to free boundary minimal submanifolds

The metrics induced on free boundary minimal surfaces in geodesic balls in the upper unit hemisphere and hyperbolic space can be characterized as critical metrics for the functionals $Θ_{r,i}$ and $Ω_{r,i}$, introduced recently by Lima, Menezes and the second author. In this paper, we generalize this characterization to free boundary minimal submanifolds of higher dimension in the same spaces. We also introduce some functionals of the form different from $Θ_{r,i}$ and show that the critical metrics for them are the metrics induced by free boundary minimal immersions into a geodesic ball in the upper unit hemisphere. In the case of surfaces, these functionals are bounded from above and not bounded from below. Moreover, the canonical metric on a geodesic disk in a 3-ball in the upper unit hemisphere is maximal for this functional on the set of all Riemannian metric of the topological disk.

math.DG↗

arXiv2025

On free boundary minimal submanifolds in geodesic balls in $\mathbb H^n$ and $\mathbb S^n_+$

We consider free boundary minimal submanifolds in geodesic balls in the hyperbolic space $\mathbb H^n$ and in the round upper hemisphere $\mathbb S^n_+$. Recently, Lima and Menezes have found a connection between free boundary minimal surfaces in geodesic balls in $\mathbb S^n_+$ and maximal metrics for a functional, defined on the set of Riemannian metrics on a given compact surface with boundary. This connection is similar to the connection between free boundary minimal submanifolds in Euclidean balls and the critical metrics of the functional "the $k$-th normalized Steklov eigenvalue", introduced by Faser and Schoen. We define two natural functionals on the set of Riemannian metrics on a compact surface with boundary. One of these functionals is the high order generalization of the functional, introduced by Lima and Menezes. We prove that the critical metrics for these functionals arise as metrics induced by free boundary minimal immersions in geodesic balls in $\mathbb H^n$ and in $\mathbb S^n_+$, respectively. We also prove a converse statement. Besides that, we discuss the (Morse) index of free boundary minimal submanifolds in geodesic balls in $\mathbb H^n$ or $\mathbb S^n_+$. We show that the index of the critical spherical catenoid in a geodesic ball in $\mathbb S^3_+$ is 4 and the index of the critical spherical catenoid in a geodesic ball in $\mathbb H^3$ is at least 4. We prove that the index of a geodesic $k$-ball in a geodesic $n$-ball in $\mathbb H^n$ or $\mathbb S^n_+$ is $n-k$. For the proof of these statements we introduce the notion of spectral index similarly to the case of free boundary minimal submanifolds in a unit ball in the Euclidean space.

math.DG↗

arXiv2023

On the Index of Fraser-Sargent-type minimal surfaces

Fraser-Sargent surfaces are free boundary minimal surfaces in the four-dimensional unit Euclidean ball. Extended infinitely they define immersed minimal surfaces in the Euclidean space. In the present paper we compute the Morse index and the nullity of these extended minimal surfaces. The parts of these surfaces outside the ball are exterior free boundary minimal surfaces. We provide a numerical evidence that they are stable. As a corollary of these results we obtain a lower bound on the index of Fraser-Sargent surfaces inside the ball. The obtained lower bound is not sharp. We provide computational experiments and state a conjecture about an improved index lower bound. Independently of it we also find an upper bound on the index of Fraser-Sargent surfaces inside the ball.

math.DG↗

arXiv2022

On the index of the critical Möbius band in $\mathbb B^4$

In this paper we prove that the Morse index of the critical Möbius band in the $4-$dimensional Euclidean ball $\mathbb B^4$ equals 5. It is conjectured that this is the only embedded non-orientable free boundary minimal surface of index 5 in $\mathbb B^4$. One of the ingredients in the proof is a comparison theorem between the spectral index of the Steklov problem and the energy index. The latter also enables us to give another proof of the well-known result that the index of the critical catenoid in $\mathbb B^3$ equals 4.

math.DG↗

arXiv2021

Degenerating sequences of conformal classes and the conformal Steklov spectrum

Let $Σ$ be a compact surface with boundary. For a given conformal class $c$ on $Σ$ the functional $σ_k^*(Σ,c)$ is defined as the supremum of the $k-$th normalized Steklov eigenvalue over all metrics on $c$. We consider the behaviour of this functional on the moduli space of conformal classes on $Σ$. A precise formula for the limit of $σ_k^*(Σ,c_n)$ when the sequence $\{c_n\}$ degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander-Nadirashvili invariants of closed manifolds defined as $\inf_{c}σ_k^*(Σ,c)$, where the infimum is taken over all conformal classes $c$ on $Σ$. We show that these quantities are equal to $2πk$ for any surface with boundary. As an application of our techniques we obtain new estimates on the $k-$th normalized Steklov eigenvalue of a non-orientable surface in terms of its genus and the number of boundary components.

math.DG↗

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↗

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↗