arXiv research

The paper finds the minimum number of negative eigenvalues for conformal Laplacian metrics.

Lower bound for Steklov eigenvalues on negatively curved manifolds.

We show that zero is not an eigenvalue of the conformal Laplacian for generic Riemannian metrics. We also discuss non-compactness for sequences of metrics with growing number of negative eigenvalues of the conformal Laplacian.

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…

Extending our previous work on eigenvalues of closed surfaces and work of Otal and Rosas, we show that a complete Riemannian surface S of finite type and negative Euler characteristic has at most negative of the Euler characteristics many small eigenvalues.

Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.

Study sharp lower bounds on negative eigenvalues of magnetic Pauli operator.

Algorithm transforms weakly negative plumbing trees to negative definite ones.

New bounds on Bartnik mass for surfaces with non-negative first eigenvalue.

CR Paneitz operator on non-embeddable tori has infinitely many negative eigenvalues

We prove a lower bound for the number of negative eigenvalues for a Schrödinger operator on a Riemannian manifold via the integral of the potential.

In this paper we consider a domain in a space of negative constant sectional curvature. Such assumption about the sectional curvature let us develop a new technique and improve existing lower bounds of eigenvalues from Dirichlet eigenvalue problem, obtained by Alessandro Savo in 2009.

The paper generalizes a Steklov eigenvalue inequality for substatic triples under non-negative Ricci curvature.

Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.

Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.

Lower bounds for eigenvalues on manifolds with negative Ricci curvature.

Lower bounds for eigenvalues on manifolds with negative Ricci curvature.

Rough and Hodge Laplacians eigenvalues approach zero with fixed volume.

We complete the picture of sharp eigenvalue estimates for the p-Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator $Δ_p$ when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and p…

In this paper we get a necessary and sufficient condition for the Ricci operator of a solvable metric Lie algebra to have at least two negative eigenvalues. In particular, this condition implies that the Ricci operator of every non-unimodular solvable metric Lie algebra or every non-abelian nilpotent metric Lie algebra…

Along the line of the Yang Conjecture, we give a new estimate on the lower bound of the first non-zero eigenvalue of a closed Riemannian manifold with negative lower bound of Ricci curvature in terms of the in-diameter and the lower bound of Ricci curvature.

Lower bound found for Steklov eigenvalue on curved manifolds.

Sharp lower bound for $p$-Laplacian eigenvalue on non-compact manifolds.

The loss function of deep networks is known to be non-convex but the precise nature of this nonconvexity is still an active area of research. In this work, we study the loss landscape of deep networks through the eigendecompositions of their Hessian matrix. In particular, we examine how important the negative eigenvalu…

Eigenvalue bounds for forms on warped manifolds studied.

This paper, focusing on the growth rate of the measure, gives pointwise bounds of solutions of eigenvalue equations of the Laplace-Beltrami operator on noncompact Riemannian manifolds.

Sharp lower bound found for geodesic ball eigenvalues.

We prove inequalities for Laplace eigenvalues on Riemannian manifolds generalising to higher eigenvalues two classical inequalities for the first Laplace eigenvalue - the inequality in terms of the $L^2$-norm of mean curvature, due to Reilly in 1977, and the inequality in terms of conformal volume, due to Li and Yau in…

In this paper, we establish sharp inequalities for four kinds of classical eigenvalues on a bounded domain of a Riemannian manifold. We also establish asymptotic formulas for the eigenvalues of the buckling and clamped plate problems. In addition, we give a negative answer to the Payne conjecture for the one-dimensiona…

We generalise a theorem of Engman and Abreu--Freitas on the first invariant eigenvalue of non-negatively curved $S^{1}$-invariant metrics on $\mathbb{CP}^{1}$ to general toric Kähler metrics with non-negative scalar curvature. In particular, a simple upper bound of the first non-zero invariant eigenvalue for such metri…

Upper bounds for Steklov eigenvalues on curved submanifolds.

We consider the problem of conformally deforming a metric to one with a prescribed symmetric function of the eigenvalues of the Ricci tensor, in the case of negative curvature.

Estimates graph curvature and diameter using Laplacian eigenvalues.

Negative curvature restricts the gap between the first and second eigenvalues of convex domains.

Study on CR Paneitz operator on non-embeddable CR manifolds.

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

We give a new estimate on the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature and provide a solution for a conjecture of H. C. Yang.

On 7D quaternionic contact manifolds, eigenvalue bounds imply special structure.

In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the GJMS operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal…

In this paper, we mainly investigate continuity, monotonicity and differentiability for the first eigenvalue of the $p$-Laplace operator along the Ricci flow on closed manifolds. We show that the first $p$-eigenvalue is strictly increasing and differentiable almost everywhere along the Ricci flow under some curvature a…

In this note, by extending the arguments of Ling (Illinois J. Math. 51, 853-860, 2007) to Bakry-Emery geometry, we shall give lower bounds for the first nonzero eigenvalue of the Witten-Laplacian on compact Bakry-Emery manifolds in the case that the Bakry-Emery Ricci curvature has some negative lower bounds and the man…

Constructs conformal metrics with negative curvature on manifolds with boundary.

Paper finds eigenvalue bounds for hyperbolic space domains.

The study finds a metric that maximizes the second eigenvalue of the Conformal Laplacian.

The paper improves eigenvalue estimates for manifolds with Ricci curvature conditions.

Minimal surfaces in spheres have unique energy properties.

In this paper we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with boundary. We prove its solvability and the compactness of the solution set, provided the Ricci tensor is non-negative definite.

Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of a bounded domain in N are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklo…