arXiv research

Researchers develop weighted GJMS operators for smooth metric measure spaces.

Study constructs transverse metrics using transformations commuting with elliptic operators.

Defines operations in non-Archimedean metrics theory.

Graph Laplace operators uniquely identify metrics and densities on manifolds.

Develops calculus for QFB metrics, proving Fredholm properties and decay of harmonic forms.

Study examines metrics and functionals for Hodge-Dirac operator on manifolds.

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…

We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption …

Sharp estimates derived for quasilinear equations on metric measure spaces.

We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator…

Study linear differential operators on special manifolds.

In this paper we discuss the index problem for geometric differential operators (Spin-Dirac operator, Gauß-Bonnet operator, Signature operator) on manifolds with metric horns. On singular manifolds these operators in general do not have unique closed extensions. But there always exist two extremal extensions $D_{min}$ …

The study examines conditions for scalar curvature and Dirac operators on singular spaces.

Unified positive mass theorem and Dirac operator study on weighted manifolds.

Derives GJMS operators and Q-curvatures for submanifolds.

Study geometric isomorphisms between spacetime solutions using paracausal metrics.

New characterization of Riemannian metric positivity and $L^2$ estimates for $d$ operator.

The aim of this paper is the study of the geodesic distance in operator groups with several Riemannian metrics. More precisely we study the geodesic distance in self-adjoint operator groups with the left invariant Riemannian metric induced by the infinite trace and extend known results about the completeness of some cl…

Study examines the geometry of a group of Fourier-integral operators related to Diff(S^1).

Paper shows how scattering maps of Schrödinger equations relate to metrics.

Study on metrics maximizing eigenvalues of Paneitz operator on 4-manifolds.

Researchers create compatibility complexes for Einstein metrics.

We confirm a conjecture of Hamilton: On compact manifolds the normalized Ricci flow evolves metrics with positive curvature operators to limit metrics with constant curvature.

A new metric compares dynamical systems using operator eigenvalues.

We prove that for cobordant closed spin manifolds of dimension $n\geq 3$ the associated spaces of metrics with invertible Dirac operator are homotopy equivalent. This is the spinorial counterpart of a similar result on positive scalar curvature of Chernysh/Walsh and generalizes the surgery result of Ammann-Dahl-Humbert…

For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result of this paper is that these properties of a metric can be preserved when the met…

Most of the 50-year history of the study of the set of knot concordance classes, C, has focused on its structure as an abelian group. Here we take a different approach, namely we study C as a metric space admitting many natural geometric operators, especially satellite operators. We consider several knot concordance sp…

Study on 3D Lie groups finds all generalized Einstein metrics.

Variational methods yield formulas for eigenvalues of elliptic operators, with applications to metric evolution.

Numerical experiments support conjecture about opers and nonabelian Hodge.

We establish the existence of analytic curves of eigenvalues for the Laplace-Neumann operator through an analytic variation of the metric of a compact Riemannian manifold $M$ with boundary by means of a new approach rather than Kato's method for unbounded operators. We obtain an expression for the derivative of the cur…

Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the fam…

In a conformal class of metrics with positive Yamabe invariant, we derive a necessary and sufficient condition for the existence of metrics with positive Q curvature. The condition is conformally invariant. We also prove some inequalities between the Green's functions of the conformal Laplacian operator and the Paneitz…

Proves spectral simplicity of Hodge Laplacian and curl operator along metric families.

Teaches Dirac operators for geometry and topology.

The article deals with intrinsic metrics, Dirac operators and spectral triples induced by regular Dirichlet and resistance forms. We show, in particular, that if a local resistance form is given and the space is compact in resistance metric, then the intrinsic metric yields a geodesic space. Given a regular Dirichlet f…

We provide an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The construction is based on results of A. Brudnyi and Yu. Brudnyi concerning linear e…

Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.

We obtain several essential self-adjointness conditions for a Schroedinger type operator D*D+V acting in sections of a vector bundle over a manifold M. Here V is a locally square-integrable bundle map. Our conditions are expressed in terms of completeness of certain metrics on M; these metrics are naturally associated …

Let (M^n, g) be a closed smooth Riemannian spin manifold and denote by D its Atiyah-Singer-Dirac operator. We study the variation of Riemannian metrics for the zeta function and functional determinant of D^2, and prove finiteness of the Morse index at stationary metrics, and local extremality at such metrics under gene…

Study reveals how boundary wave operator determines metric properties on anti-de Sitter spacetimes.

We generalize the result of [Matveev-Topalov 2001] to all signatures: we show that in all signatures the Killing tensors constructed by projectively equivalent metrics correspond to commuting differential operators

A new method matches similar regions in non-rigid shapes using spectra of differential operators.

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigen…

On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended over the trace of a surgery of codimension at least three. We then prove that if non-empty the space of metrics with inver…

New criterion for wave operators on Kato-Ricci manifolds.

This is a survey article on a known generalization of Dirac-type operators to transverse operators called basic Dirac operators on Riemannian foliations, which are smooth foliations that have a transverse geometric structure. Construction of these operators requires the additional structure of what is called a bundle-l…

New Kähler manifolds found with nonpositive curvature operators.