arXiv research

Sharp eigenvalue bounds and splitting for modified Ricci flow.

We study the Dirac spectrum on compact Riemannian spin manifolds $M$ equipped with a metric connection $\nabla$ with skew torsion $T\inΛ^3 M$ in the situation where the tangent bundle splits under the holonomy of $\nabla$ and the torsion of $\nabla$ is of `split' type. We prove an optimal lower bound for the first eige…

Minimal Morse functions on Poincaré dodecahedral space are selected via spectral properties.

Study compares manifolds with boundary under weighted Ricci curvature bounds.

It is proved that a compact Kahler manifold whose Ricci tensor has two distinct, constant, non-negative eigenvalues is locally the product of two Kahler-Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Irreducible Kahler manifolds with two distinct, constant eigenvalues of the Ricci…

A universal lower bound for the first positive eigenvalue of the Dirac operator on a compact quaternionic Kaehler manifold M of positive scalar curvature is calculated. It is shown that it is equal to the first positive eigenvalue on the quaternionic projective space. For this, the horizontal tangent bundle on the cano…

In this paper, we establish a kind of splitting theorem for the eigenvalues of a specific family of operators on the base of a warped product. As a consequence, we prove a density theorem for a set of warping functions that makes the spectrum of the Laplacian a warped-simple spectrum. This is then used to study the gen…

We study the (massless) Dirac operator on a 3-sphere equipped with Riemannian metric. For the standard metric the spectrum is known. In particular, the eigenvalues closest to zero are the two double eigenvalues +3/2 and -3/2. Our aim is to analyse the behaviour of eigenvalues when the metric is perturbed in an arbitrar…

The paper proves conditions for Kähler-Einstein metrics on certain bundles.

We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: in particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take th…

Analyzes tunneling effects for Schrödinger operators on vector bundles.

The paper proves that most metrics satisfy a strong version of Arnold's conjecture for Laplace eigenvalues.

Optimal data splitting improves covariance matrix estimation in large datasets.

Two metrics $g$ and $\bar g$ are geodesically equivalent, if they share the same (unparameterized) geodesics. We introduce two constructions that allow one to reduce many natural problems related to geodesically equivalent metrics, such as the classification of local normal forms and the Lie problem (the description o…

Paper analyzes holdout cross-validation for large non-Gaussian covariance estimation.

We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed radii, a volume growth rigidity theorem for the metric neighborhoods of the boundar…

We examine three key conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b_1 conjecture and the virtually fibred conjecture. We explore the interaction of these conjectures with the following seemingly unrelated areas: eigenvalues of the Laplacian, and Heegaard splittings. We first gi…

We study Riemannian manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $N$ at most $1$, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with boundary. We conclude a volume growth rigidity theorem for the metric neighborhoods of …

There are two themes in the present paper. The first one is spelled out in the title, and is inspired by an attempt to find an analogue of Hersch-Yang-Yau estimate for $lambda_1$ of surfaces in symplectic category. In particular we prove that every split symplectic manifold $T^4 times M$ admits a compatible Riemannian …

We formulate a more conceptual interpretation of the Cappell-Lee-Miller glueing/splitting theorem using the new language of asymptotic maps and asymptotic exactness. Additionally, we present an asymptotic description of the Mayer-Vietoris sequence naturally associated to the Cech cohomology of the sheaf of local soluti…

New theorem on flat tori stability using harmonic maps and Ricci flow.

Researchers compute the spectrum of Hodge-Laplacian on 1-forms for SU(2) and SO(3).

Test partial effects in Frechet regression on Bures-Wasserstein manifolds.

We study hamiltonian actions of compact groups in the presence of compatible involutions. We show that the lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to involutions on the group strongly inner to the given one. Our …

Researchers describe local properties of Haantjes operators.

We prove that a compact $RCD^*(0,N)$ (or equivalently $RCD(0,N)$) metric measure space, $\left(X, d, m \right)$, with $\diam X \le d$ and its first (nonzero) eigenvalue of the Laplacian (in the sense of Ambrosio-Gigli-Savaré) , $λ_1 = \frac{π^2}{d^2}$, has to be a circle or a line segment with diameter, $π$. This compl…

In this paper, we investigate complete Riemannian manifolds satisfying the lower weighted Ricci curvature bound $\mathrm{Ric}_{N} \geq K$ with $K>0$ for the negative effective dimension $N<0$. We analyze two $1$-dimensional examples of constant curvature $\mathrm{Ric}_N \equiv K$ with finite and infinite total volumes.…

I In this paper, first we study a complete smooth metric measure space $(M^n,g, e^{-f}dv)$ with the ($\infty$)-Bakry-Émery Ricci curvature $\textrm{Ric}_f\ge \frac a2g$ for some positive constant $a$. It is known that the spectrum of the drifted Laplacian $Δ_f$ for $M$ is discrete and the first nonzero eigenvalue of $Δ…

The paper studies Nijenhuis operators with a unity and their connection to F-manifolds.

We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational, integer, and finite field coefficients, the hyperbolic volume …

In this paper, we consider a Riemannian manifold (M, g) endowed with a Riemannian flow and we study the curvature term in the Bochner-Weitzenb{ö}ck formula of the basic Laplacian on M. We prove that this term splits into two parts. The first part depends mainly on the curvature operator of the underlying manifold M and…

This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also prov…

The paper compares Steklov and Laplacian eigenvalues on graphs.

Memristors have recently received significant attention as ubiquitous device-level components for building a novel generation of computing systems. These devices have many promising features, such as non-volatility, low power consumption, high density, and excellent scalability. The ability to control and modify biasin…

Paper uses Random Matrix Theory for optimal training-testing data split.

In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem of the weighted La…

The paper explores inequalities between eigenvalues on Riemannian manifolds.

Eigenvalues of Steklov eigenproblems change predictably with boundary tweaks.

Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.

The paper sets lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.

Paper finds how Steklov eigenvalues change on graphs and trees.

Eigenvalue estimate for shrinkers in mean curvature flow.

In this paper we study eigenvalues of the closed eigenvalue problem of the Witten-Laplacian on an $n$-dimensional compact Riemannian manifold. Estimates for eigenvalues are given. As applications, we give a sharp upper bound for the $k^{\text{th}}$ eigenvalue and for isoparametric minimal hypersurfaces in the unit sphe…

For a bounded domain $Ω$ with a piecewise smooth boundary in an $n$-dimensional Euclidean space $\mathbf{R}^{n}$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for eigenvalues of the Laplacian. As an application, we study lower order eigenvalues of the Lap…

The paper provides estimates for eigenvalues of elliptic differential problems.

The paper compares eigenvalues of Dirichlet, Neumann, and Laplacian on graphs.

Sharp bounds derived for the first two Steklov eigenvalues of exterior domains.

SPlit optimizes dataset splitting for better model performance.