A note on Kuttler-Sigillito's inequalities

We provide several inequalities between eigenvalues of some classical eigenvalue problems on domains with $C^2$ boundary in complete Riemannian manifolds. A key tool in the proof is the generalized Rellich identity on a Riemannian manifold. Our results in particular extend some inequalities due to Kutller and Sigillito from subsets of $\mathbb{R}^2$ to the manifold setting

Sub-Laplacian eigenvalue bounds on sub-Riemannian manifolds

We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues λk of conformal sub-Riemannian metrics that are asymptotically sharp as k→+∞. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for contact metric manifolds conformal to such Sasakian manifolds, we obtain eigenvalue inequalities that can be viewed as versions of the classical results by Korevaar and Buser in Riemannian geometry

Higher order Cheeger inequalities for Steklov eigenvalues

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Steklov eigenvalues. In particular it extends the Cheeger type inequality for the first nonzero Steklov eigenvalue previously studied by Escobar in 1997 and by Jammes in 2015 to higher order Steklov eigenvalues. The technique we develop to get this lower bound is based on considering a family of accelerated Markov operators in the finite and mesurable situations and of mass concentration deformations of the Laplace-Beltrami operator in the manifold setting which converges uniformly to the Steklov operator. As an intermediary step in the proof of the higher order Cheeger type inequality, we define the Dirichlet-Steklov connectivity spectrum and show that the Dirichlet connectivity spectra of this family of operators converges to (or bounded by) the Dirichlet-Steklov spectrum uniformly. Moreover, we obtain bounds for the Steklov eigenvalues in terms of its Dirichlet-Steklov connectivity spectrum which is interesting in its own right and is more robust than the higher order Cheeger type inequalities. The Dirichlet-Steklov spectrum is closely related to the Cheeger-Steklov constants

Escobar constants of planar domains

We initiate the study of the higher-order Escobar constants $I_k(M)$, $k\geq 3$, on bounded planar domains $M$. The Escobar constants $I_k$ of the unit disk and a family of polygons are provided

Eigenvalue inequalities on Riemannian manifolds with a lower Ricci curvature bound

We revisit classical eigenvalue inequalities due to Buser, Cheng, and Gromov on closed Riemannian manifolds, and prove the versions of these results for the Dirichlet and Neumann boundary value problems. Eigenvalue multiplicity bounds and related open problems are also discussed

Eigenvalue bounds of mixed Steklov problems

. The Steklov–Neumann eigenvalue problem is also called the sloshing problem. We obtain two-term asymptotically sharp lower bounds on the Riesz means of the sloshing problem and also provide an asymptotically sharp upper bound for the Riesz means of mixed Steklov–Dirichlet problem. The proof of our results for the sloshing problem uses the average variational principle and monotonicity of sloshing eigenvalues. In the case of Steklov–Dirichlet eigenvalue problem, the proof is based on a well-known bound on the Riesz means of the Dirichlet fractional Laplacian, and an inequality between the Dirichlet and Navier fractional Laplacian. The two-term asymptotic results for the Riesz means of mixed Steklov eigenvalue problems are discussed in the Appendix which in particular show the asymptotic sharpness of the bounds we obtain

Spectral geometry of the Steklov problem on orbifolds

We consider how the geometry and topology of a compact $n$-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in terms of only boundary data. As a consequence, we prove that the Steklov spectrum detects the presence and number of orbifold singularities on the boundary of an orbisurface and it detects the number each of smooth and singular boundary components. Moreover, we find that the Steklov spectrum also determines the lengths of the boundary components modulo an equivalence relation, and we show by examples that this result is the best possible. We construct various examples of Steklov isospectral Riemannian orbifolds which demonstrate that these two-dimensional results do not extend to higher dimensions. In addition, we give two- imensional examples which show that the Steklov spectrum does \emph{not} detect the presence of interior singularities nor does it determine the orbifold Euler characteristic. In fact, a flat disk is Steklov isospectral to a cone. In another direction, we obtain upper bounds on the Steklov eigenvalues of a Riemannian orbifold in terms of the isoperimetric ratio and a conformal invariant. We generalize results of B. Colbois, A. El Soufi and A. Girouard, and the fourth author to the orbifold setting; in the process, we gain a sharpness result on these bounds that was not evident in the manifold setting. In dimension two, our eigenvalue bounds are solely in terms of the orbifold Euler characteristic and the number each of smooth and singular boundary components

Neotethyan Subduction Ignited the Iran Arc and Backarc Differently

Most arcs show systematic temporal and spatial variations in magmatism with clear shifts in igneous rock compositions between those of the magmatic front (MF) and those in the backarc (BA). It is unclear if similar magmatic polarity is seen for extensional continental arcs. Herein, we use geochemical and isotopic characteristics coupled with zircon U‐Pb geochronology to identify the different magmatic style of the Iran convergent margin, an extensional system that evolved over 100 Myr. Our new and compiled U‐Pb ages indicate that major magmatic episodes for the NE Iran BA occurred at 110–80, 75–50, 50–35, 35–20, and 15–10 Ma. In contrast to NE Iran BA magmatic episodes, compiled data from MF display two main magmatic episodes at 95–75 and 55–5 Ma, indicating more continuous magmatism for the MF than for the BA. We show that Paleogene Iran serves as a useful example of a continental arc under extension. Our data also suggest that there is not a clear relationship between the subduction velocity of Neotethyan Ocean beneath Iran and magmatic activity in Iran. Our results imply that the isotopic compositions of Iran BA igneous rocks do not directly correspond to the changes in tectonic processes or geodynamics, but other parameters such as the composition of lithosphere and melt source(s) should be considered. In addition, changes in subduction zone dynamics and contractional versus extensional tectonic regimes influenced the composition of MF and BA magmatic rocks. These controls diminished the geochemical and isotopic variations between the magmatic front and backarc

Participatory systems mapping for population health research, policy and practice: guidance on method choice and design

Executive Summary: What is participatory systems mapping? Participatory systems mapping engages stakeholders with varied knowledge and perspectives in creating a visual representation of a complex system. Its purpose is to explore, and document perceived causal relations between elements in the system. This guidance focuses on six causal systems mapping methods: systems-based theory of change maps; causal loop diagrams; CECAN participatory systems mapping; fuzzy cognitive maps; systems dynamics models; and Bayesian belief networks. What is the purpose of this guidance? This guidance includes a Framework that aids the choice and design of participatory systems mapping approaches for population health research, policy and practice. It offers insights on different systems mapping approaches, by comparing them and highlighting their applications in the population health domain. This guidance also includes case studies, signposting to further reading and resources, and recommendations on enhancing stakeholder involvement in systems mapping. Who is this guidance for? This guidance is designed for anyone interested in using participatory systems mapping, regardless of prior knowledge or experience. It primarily responds to calls to support the growing demand for systems mapping (and systems-informed approaches more broadly) in population health research, policy and practice. This guidance can however also be applied to other disciplines. How was it developed? The guidance was created by an interdisciplinary research team through an iterative, rigorous fivestage process that included a scoping review, key informant interviews, and a consultation exercise with subject experts. What is the ‘Participatory Systems Design Framework’ included in this guidance? The Design Framework supports users to choose between different methods and enhance the design of participatory systems mapping projects. Specifically, it encourages users to consider: 1) the added value of adopting a participatory approach to systems mapping; 2) the differences between methods, including their relative advantages and disadvantages; and 3) the feasibility of using particular methods for a given purpose. An editable version of the Framework is available to download as a supplementary file. How will this guidance support future use of these methods? Participatory systems mapping is an exciting and evolving field. This guidance clarifies and defines the use of these methods in population health research, policy and practice, to encourage more thoughtful and purposeful project design, implementation, and reporting. The guidance also identifies several aspects for future research and development: methodological advancements; advocating for and strengthening participatory approaches; strengthening reporting; understanding and demonstrating the use of maps; and developing skills for the design and use of these methods

Eigenvalues of the Laplacian and extrinsic geometry

We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second, we consider complex submanifolds of the complex projective space $\mathbb{C} P^N$ instead of submanifolds of $\mathbb{R}^N$ and we obtain an eigenvalue upper bound depending only on the dimension of the submanifold which is sharp for the first non-zero eigenvalue