Professor Norbert Peyerimhoff's Outputs (70)

Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature (2020)
Journal Article
Cushing, D., Kamtue, S., Liu, S., Muench, F., & Peyerimhoff, N. (2020). Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature. Advances in Mathematics, 360, Article 107188. https://doi.org/10.1016/j.aim.2020.107188

We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the Ollivier Ricci curvature sense. Our main result is a classification of all self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs, even-dimensional demi-cub... Read More about Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature.

Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces (2020)
Journal Article
Knieper, G., Parker, J. R., & Peyerimhoff, N. (2020). Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces. Differential Geometry and its Applications, 69, Article 101605. https://doi.org/10.1016/j.difgeo.2020.101605

In this article we consider solvable hypersurfaces of the form with induced metrics in the symmetric space , where H a suitable unit length vector in the subgroup A of the Iwasawa decomposition . Since M is rank 2, A is 2-dimensional and we can param... Read More about Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces.

Quartic graphs which are Bakry-Émery curvature sharp (2019)
Journal Article
Cushing, D., Kamtue, S., Peyerimhoff, N., & Watson May, L. (2020). Quartic graphs which are Bakry-Émery curvature sharp. Discrete Mathematics, 343(3), Article 111767. https://doi.org/10.1016/j.disc.2019.111767

We give a classification of all connected quartic graphs which are (infinity) curvature sharp in all vertices with respect to Bakry-Émery curvature. The result is based on a computer classification by F. Gurr and L. Watson May and a combinatorial cas... Read More about Quartic graphs which are Bakry-Émery curvature sharp.

The Fourier Transform on harmonic manifolds of purely exponential volume growth (2019)
Journal Article
Biswas, K., Knieper, G., & Peyerimhoff, N. (2021). The Fourier Transform on harmonic manifolds of purely exponential volume growth. Journal of Geometric Analysis, 31(1), 126-163. https://doi.org/10.1007/s12220-019-00253-9

Let X be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of non-compact harmonic manifolds except... Read More about The Fourier Transform on harmonic manifolds of purely exponential volume growth.

Trivalent expanders, $(Delta – Y)$-transformation, and hyperbolic surfaces (2019)
Journal Article
Ivrissimtzis, I., Peyerimhoff, N., & Vdovina, A. (2019). Trivalent expanders, $(Delta – Y)$-transformation, and hyperbolic surfaces. Groups, Geometry, and Dynamics, 13(3), 1103-1131. https://doi.org/10.4171/ggd/518

We construct a new family of trivalent expanders tessellating hyperbolic surfaces with large isometry groups. These graphs are obtained from a family of Cayley graphs of nilpotent groups via (Delta–Y)-transformations. We study combinatorial, topologi... Read More about Trivalent expanders, $(Delta – Y)$-transformation, and hyperbolic surfaces.

A support theorem for the X-ray transform on manifolds with plane covers (2019)
Journal Article
Peyerimhoff, N., & Samiou, E. (2020). A support theorem for the X-ray transform on manifolds with plane covers. Mathematical Proceedings of the Cambridge Philosophical Society, 169(1), 149-158. https://doi.org/10.1017/s0305004119000148

This paper is concerned with support theorems of the X-ray transform on non-compact manifolds with conjugate points. In particular, we prove that all simply connected 2-step nilpotent Lie groups have a support theorem. Important ingredients of the pr... Read More about A support theorem for the X-ray transform on manifolds with plane covers.

Distance Bounds for Graphs with Some Negative Bakry-Émery Curvature (2019)
Journal Article
Liu, S., Münch, F., Peyerimhoff, N., & Rose, C. (2019). Distance Bounds for Graphs with Some Negative Bakry-Émery Curvature. Analysis and Geometry in Metric Spaces, 7(1), 1-14. https://doi.org/10.1515/agms-2019-0001

We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit... Read More about Distance Bounds for Graphs with Some Negative Bakry-Émery Curvature.

Curvature and higher order Buser inequalities for the graph connection Laplacian (2019)
Journal Article
Liu, S., Muench, F., & Peyerimhoff, N. (2019). Curvature and higher order Buser inequalities for the graph connection Laplacian. SIAM Journal on Discrete Mathematics, 33(1), 257-305. https://doi.org/10.1137/16m1056353

We study the eigenvalues of the connection Laplacian on a graph with an orthogonal group or unitary group signature. We establish higher order Buser type inequalities, i.e., we provide upper bounds for eigenvalues in terms of Cheeger constants in the... Read More about Curvature and higher order Buser inequalities for the graph connection Laplacian.

Minimizing length of billiard trajectories in hyperbolic polygons (2018)
Journal Article
Parker, J. R., Peyerimhoff, N., & Siburg, K. F. (2018). Minimizing length of billiard trajectories in hyperbolic polygons. Conformal Geometry and Dynamics, 22, 315-332. https://doi.org/10.1090/ecgd/328

Closed billiard trajectories in a polygon in the hyperbolic plane can be coded by the order in which they hit the sides of the polygon. In this paper, we consider the average length of cyclically related closed billiard trajectories in ideal hyperbol... Read More about Minimizing length of billiard trajectories in hyperbolic polygons.

Bakry-Émery Curvature Functions on Graphs (2018)
Journal Article
Cushing, D., Liu, S., & Peyerimhoff, N. (2020). Bakry-Émery Curvature Functions on Graphs. Canadian Journal of Mathematics, 72(1), 89-143. https://doi.org/10.4153/cjm-2018-015-4

We study local properties of the Bakry-Émery curvature function KG,x:(0,∞]→R at a vertex x of a graph G systematically. Here KG,x(N) is defined as the optimal curvature lower bound K in the Bakry-Émery curvature-dimension inequality CD(K,N) that x sa... Read More about Bakry-Émery Curvature Functions on Graphs.

Ollivier-Ricci idleness functions of graphs (2018)
Journal Article
Bourne, D., Cushing, D., Liu, S., Muench, F., & Peyerimhoff, N. (2018). Ollivier-Ricci idleness functions of graphs. SIAM Journal on Discrete Mathematics, 32(2), 1408-1424. https://doi.org/10.1137/17m1134469

We study the Ollivier--Ricci curvature of graphs as a function of the chosen idleness. We show that this idleness function is concave and piecewise linear with at most three linear parts, and at most two linear parts in the case of a regular graph. W... Read More about Ollivier-Ricci idleness functions of graphs.

Eigenvalue ratios of non-negatively curved graphs (2018)
Journal Article
Liu, S., & Peyerimhoff, N. (2018). Eigenvalue ratios of non-negatively curved graphs. Combinatorics, Probability and Computing, 27(5), 829-850. https://doi.org/10.1017/s0963548318000214

We derive an optimal eigenvalue ratio estimate for finite weighted graphs satisfying the curvature-dimension inequality CD(0, ∞). This estimate is independent of the size of the graph and provides a general method to obtain higher-order spectral esti... Read More about Eigenvalue ratios of non-negatively curved graphs.

Bakry–Émery curvature and diameter bounds on graphs (2018)
Journal Article
Liu, S., Münch, F., & Peyerimhoff, N. (2018). Bakry–Émery curvature and diameter bounds on graphs. Calculus of Variations and Partial Differential Equations, 57(2), Article 67. https://doi.org/10.1007/s00526-018-1334-x

We prove finiteness and diameter bounds for graphs having a positive Ricci-curvature bound in the Bakry–Émery sense. Our first result using only curvature and maximal vertex degree is sharp in the case of hypercubes. The second result depends on an a... Read More about Bakry–Émery curvature and diameter bounds on graphs.

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space (2017)
Journal Article
Peyerimhoff, N., Täufer, M., & Veselić, I. (2017). Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space. Nanosistemy: fizika, himiâ, matematika Наносистемы: физика, химия, математика (Print), 8(2), 216-230. https://doi.org/10.17586/2220-8054-2017-8-2-216-230

For the analysis of the Schrödinger and related equations it is of central importance whether a unique continuation principle (UCP) holds or not. In continuum Euclidean space, quantitative forms of unique continuation imply Wegner estimates and regul... Read More about Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space.

Sectional curvature of polygonal complexes with planar substructures (2016)
Journal Article
Keller, M., Peyerimhoff, N., & Pogorzelski, F. (2017). Sectional curvature of polygonal complexes with planar substructures. Advances in Mathematics, 307, 1070-1107. https://doi.org/10.1016/j.aim.2016.10.027

In this paper we introduce a class of polygonal complexes for which we consider a notion of sectional combinatorial curvature. These complexes can be viewed as generalizations of 2-dimensional Euclidean and hyperbolic buildings. We focus on the case... Read More about Sectional curvature of polygonal complexes with planar substructures.

Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians (2015)
Journal Article
Lange, C., Liu, S., Peyerimhoff, N., & Post, O. (2015). Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians. Calculus of Variations and Partial Differential Equations, 54(4), 4165-4196. https://doi.org/10.1007/s00526-015-0935-x

We discuss a Cheeger constant as a mixture of the frustration index and the expansion rate, and prove the related Cheeger inequalities and higher order Cheeger inequalities for graph Laplacians with cyclic signatures, discrete magnetic Laplacians on... Read More about Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians.

Geometric properties of rank one asymptotically harmonic manifolds (2015)
Journal Article
Knieper, G., & Peyerimhoff, N. (2015). Geometric properties of rank one asymptotically harmonic manifolds. Journal of Differential Geometry, 100(3), 507-532. https://doi.org/10.4310/jdg/1432842363

In this article we consider asymptotically harmonic manifolds which are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature h. We prove the following equivalences for... Read More about Geometric properties of rank one asymptotically harmonic manifolds.

Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials (2015)
Journal Article
Leonhardt, K., Peyerimhoff, N., Tautenhahn, M., & Veselic, I. (2015). Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials. Reviews in Mathematical Physics, 27(04), https://doi.org/10.1142/s0129055x15500075

We study Schrödinger operators on L2(ℝd) and ℓ2(ℤd) with a random potential of alloy-type. The single-site potential is assumed to be exponentially decaying but not necessarily of fixed sign. In the continuum setting, we require a generalized step-fu... Read More about Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials.

Harmonic Functions on Rank One Asymptotically Harmonic Manifolds (2015)
Journal Article
Knieper, G., & Peyerimhoff, N. (2015). Harmonic Functions on Rank One Asymptotically Harmonic Manifolds. Journal of Geometric Analysis, 26(2), 750-781. https://doi.org/10.1007/s12220-015-9570-1

Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature hh. In this article we present results for harmonic functions on rank one as... Read More about Harmonic Functions on Rank One Asymptotically Harmonic Manifolds.

An infinite family of 2-groups with mixed Beauville structures (2014)
Journal Article
Barker, N., Boston, N., Peyerimhoff, N., & Vdovina, A. (2015). An infinite family of 2-groups with mixed Beauville structures. International Mathematics Research Notices, 2015(11), 3598-3618. https://doi.org/10.1093/imrn/rnu045

We construct an infinite family of triples (Gk, Hk, Tk), where Gk are 2-groups of increasing order, Hk are index 2 subgroups of Gk, and Tk are pairs of generators of Hk. We show that the triples uk = (Gk, Hk, Tk) are mixed Beauville structures if k i... Read More about An infinite family of 2-groups with mixed Beauville structures.