arXiv research

In 1996, Nadirashvili used Runge's theorem to produce a complete minimal disc inside a ball in R^3. In this paper we generalize the techniques used by Nadirashvili to obtain new examples of complete minimal surfaces inside a ball in R^3, with the conformal structure of an annulus.

Optimizes maps and eigenvalues on manifolds.

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 …

Let $u$ be a harmonic function in the unit ball $B(0,1) \subset \mathbb{R}^n$, $n \geq 3$, such that $u(0)=0$. Nadirashvili conjectured that there exists a positive constant $c$, depending on the dimension $n$ only, such that $H^{n-1}(\{u=0 \}\cap B) \geq c$. We prove Nadirashvili's conjecture as well as its counterpar…

Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.

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 metr…

In this paper, we investigate the relationship between the discreteness of the spectrum of a non-compact, extrinsically bounded submanifold $\varphi \colon M^m \ra N^n$ and the Hausdorff dimension of its limit set $\lim\varphi$. In particular, we prove that if $\varphi \colon \!M^2 \ra D \subseteq \R^3$ is a minimal im…

We give a characterization of conformal classes realizing a compact manifold's Yamabe invariant. This characterization is the analogue of an observation of Nadirashvili for metrics realizing the maximal first eigenvalue, and of Fraser and Schoen for metrics realizing the maximal first Steklov eigenvalue.

In this paper, we settle in the affirmative the Jakobson-Levitin-Nadirashvili-Nigam-Polterovich conjecture, stating that a certain singular metric on the Bolza surface, with area normalized, should maximize the first eigenvalue of the Laplacian.

Researchers prove existence of metrics maximizing Laplace eigenvalue on all closed surfaces.

New proof confirms flat equilateral torus is λ1-maximal.

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 pose…

We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of $\mathbb{R}^{3}$ is discrete. This give…

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 Jako…

A classical result by Cheng in 1976, improved later by Besson and Nadirashvili, says that the multiplicities of the eigenvalues of the Schrodinger operator with a smooth potential on a compact Riemannian surface M are bounded in terms of the eigenvalue index and the genus of M. We prove that these multiplicity bounds h…

The paper characterizes eigenvalues of surfaces using min-max quantities for harmonic maps.

In this paper we will prove the Calabi-Yau conjectures for embedded surfaces. In fact, we will prove considerably more. The Calabi-Yau conjectures about surfaces date back to the 1960s. Much work has been done on them over the past four decades. In particular, examples of Jorge-Xavier from 1980 and Nadirashvili from 19…

Paper proves linearity of solutions to degenerate elliptic equations in 3D.

Detailed proofs and extensions of eigenvalue multiplicity bounds for spheres and plane domains.

Affirm Lord Rayleigh's conjecture on curved spaces for clamped plates.

We study the effect of two types of degeneration of the Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point with…

This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and Fefferman including their solution of the conjecture in the case of real-analytic Rieman…

We define a new differential invariant a compact manifold by $V_{\mathcal M}(M)=\inf_g V_c(M,[g])$, where $V_c(M,[g])$ is the conformal volume of $M$ for the conformal class $[g]$, and prove that it is uniformly bounded above. The main motivation is that this bound provides a upper bound of the Friedlander-Nadirashvili…

Jakobson and Nadirashvili \cite{JN} constructed a sequence of eigenfunctions on $T^2$ with a bounded number of critical points, answering in the negative the question raised by Yau \cite{Yau1} which asks that whether the number of the critical points of eigenfunctions for the Laplacian increases with the corresponding …

Study of loops in sums of Laplace eigenfunctions on surfaces.

Paper derives second variation formula for eigenvalue functionals on surfaces.

The paper proves a new method to find the maximum Laplace eigenvalues on surfaces.

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 eigenvalu…

We prove that on any compact manifold $M^n$ with boundary, there exist a conformal class $C$ such that for any riemannian metric $g\in C$, $λ_1(M^n,g)Vol(M^n,g)^{2/n}< n.Vol(S^n,g_{\textrm{can}})^{2/n}$ and $σ_1(M,g,ρ)\mathcal M(\partial M)Vol(M)^{\frac{2-n}n}<n.Vol(S^n,g_{\textrm{can}})^{2/n}$, where $λ_1(M^n,g)$ deno…

Study of Steklov eigenvalues on degenerating conformal classes.

Optimizes metrics on surfaces for eigenvalues.

Let $M$ be a compact connected manifold of dimension $n$ endowed with a conformal class $C$ of Riemannian metrics of volume one. For any integer $k\geq0$, we consider the conformal invariant $λ_k ^c (C)$ defined as the supremum of the $k$-th eigenvalue $λ_k (g)$ of the Laplace-Beltrami operator $Δ_g$, where $g$ runs ov…

New tools for studying Hsiang algebras discovered, linking them to known algebraic structures.

The paper finds rotationally symmetric critical metrics for Laplace eigenvalues on tori.

Minimal surfaces with uniform curvature (or area) bounds have been well understood and the regularity theory is complete, yet essentially nothing was known without such bounds. We discuss here the theory of embedded (i.e., without self-intersections) minimal surfaces in Euclidean 3-space without a priori bounds. The st…

New isoperimetric inequality for clamped plates in RCD(0,N) spaces, sharp and stable.

Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by $λ_k(M,g)$ the $k$-th eigenvalue of the Laplace-Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M, g)\mapsto λ_k(M,g)$. We propose a computational method for finding the conformal spectrum $Λ^c_k(M,[g_0])$, which is d…

We study Lord Rayleigh's problem for clamped plates on an arbitrary $n$-dimensional $(n\geq 2)$ Cartan-Hadamard manifold $(M,g)$ with sectional curvature $\textbf{K}\leq -κ^2$ for some $κ\geq 0.$ We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in $(M,g)$ is universally bounde…

AI generates theorems and proofs for training theorem provers.

Global inverse function theorem proved easily using Riemannian geometry.

A new comparison theorem for geometric spaces.

Paper develops formulas and theorems in Hermitian geometry.

The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.

Revises a theorem by Thurston, finding a counter-example and a weaker version.

Proofs for Moon's theorem and its generalization.

Analyzes Saito vanishing theorem using $L^{2}$ methods.

Investigates proving geometric theorems over complex and real numbers using tilings.

Extends symplectic reduction and theorem to Lie algebroids.