arXiv research

Proves Nadirashvili's conjecture and Yau's lower bound for Laplace eigenfunctions.

Review of Yau's conjecture on zero sets of Laplace eigenfunctions.

Study on Friedlander-Nadirashvili invariants on surfaces, especially non-orientable ones.

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…

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

Paper proves a metric on a genus two surface maximizes Laplacian eigenvalue.

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…

Maximizes Laplace eigenvalues on a sphere, proving a 2002 conjecture.

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.

The paper generalizes a theorem about metrics realizing maxima of the first non-zero Laplace eigenvalue.

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.

Study of loops in sums of Laplace eigenfunctions on surfaces.

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

New proof confirms flat equilateral torus is λ1-maximal.

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 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…

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

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…

Paper solves isoperimetric inequalities and improves bounds on minimal spheres.

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

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 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…

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 fundamental tones of clamped plates on curved spaces.

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.

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

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

Paper derives second variation formula for eigenvalue functionals on surfaces.

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

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.

Polyhedra volume conjecture supports Stoker conjecture weakly.

Survey on two non-Kähler geometry conjectures.

The paper generalizes a surgery conjecture and proves it under the Zilber-Pink conjecture.

Numerical study confirms Brennan's conjecture for a counterexample to Thurston's $K=2$ conjecture.

The non-vanishing conjecture implies the abundance conjecture in certain cases.

Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equi…

The Burghelea conjecture is proven for many groups, but not all, with counter-examples provided.

Paper discusses conjectures and proves some related inequalities.

Symmetry-breaking in three differential geometry conjectures.