The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph. Some new results and generalizations are added. † This article appeared in “Graph Theory, Combinatorics, and Applications”, Vol. 2, Ed. Y. Alavi, G. Chartrand, O. R. Oellermann, A. J. Schwenk, Wiley, 1991, pp. 871–898. ‡ The work supported in part by the Research Council of Slovenia, Yugoslavia. Part of the work was done while the author was a Fulbright Scholar at the Ohio State University, Columbus, Ohio.

The Laplacian spectrum of graphs

Introduction BASIC FACTS ON HARMONIC MORPHISMS 1. Complex-valued harmonic morphisms on three-dimensional Euclidean space 2. Riemannian manifolds and conformality 3. Harmonic mappings between Riemannian manifolds 4. Fundamental properties of harmonic morphisms 5. Harmonic morphisms defined by polynomials TWISTOR METHODS 6. Mini-twistor theory on three-dimensional space-forms 7. Twistor methods 8. Holomorphic harmonic morphisms 9. Multivalued harmonic morphisms TOPOLOGICAL AND CURVATURE CONSIDERATIONS 10. Harmonic morphisms from compact 3-manifolds 11. Curvature considerations 12. Harmonic morphisms with one-dimensional fibres 13. Reduction techniques FURTHER DEVELOPMENTS 14. Harmonic morphisms between semi-Riemannian manifolds Appendix Glossary of Notation Bibliography Index

Harmonic morphisms between Riemannian manifolds

(1.1) Some of the main results described in [Report] are the following (in rough terms; notations and precise references will be given below): (1) A map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\\ is harmonic if and only if it is a critical point of the energy functional. (2) Let (M, g) and (N, h) be compact, and <̂ 0: (M, g) -> (N, h) a map. Then ^0 can be deformed to a harmonic map with minimum energy in its homotopy class in the following cases: (a) Riem ' f t ^0; (b) dim M = 2 and n2(N) = 0. (3) Any map 0O: S m -> S can be deformed to a harmonic map provided m ^ 7. More generally, suitably restricted harmonic polynomial maps can be joined to provide harmonic maps between spheres. (4) The homotopy class of maps of degree 1 from the 2-torus T to the 2-sphere S has no harmonic representative, whatever Riemannian metrics are put on T and S. (5) If in (2) M has a smooth boundary, then various Dirichlet problems have solutions in case (a) and (b); and also when the boundary data is sufficiently small.

Another Report on Harmonic Maps

Can one hear the shape of a drum

https://people.reed.edu/~davidp/412/handouts/rr.pdf

Riemann–Roch and Abel–Jacobi theory on a finite graph

Introduction. In this paper, we discuss generic properties of the eigenvalues of the Laplacian for compact Riemannian manifolds without boundary. Throughout this paper, let M be an arbitrary fixed connected compact C°° manifold of dimension n without boundary, and ^t the set of all C°° Riemannian metrics on M. For g e^t, let Δg be the Laplacian (cf. (2.1)) of (M, g) acting on the space C°°(M) of all C°° real valued functions on M and 0 = λo(flf) < λΛflO ^ x2(g) ^ T°° the eigenvalues of the Laplacian Δg counted with their multiplicities. We regard each eigenvalue Xk(g), k = 0,1, 2, , as a function of g in ^. Let us consider the following problem: "Does each eigenvalue Xk(g) depend continuously on g in ^t with respect to the C°° topology*!" The continuous dependence of the eigenvalues of the Dirichlet problem upon variations of domains is well known (cf. [CH, p. 290]). Variations of coefficients of elliptic differential operators were dealt with by KodairaSpencer [KS] who gave a proof of the continuity of eigenvalues. In this paper, we give a simple proof of the above problem. To answer the above problem, in § 1, we introduce a complete distance p on ^t which gives the C°° topology. Then, in § 2, we assert that each χk(g)f k = 1, 2, , depends continuously on g e ^t with respect to the topology on Λ? induced by the distance p. More precisely, we have

/pdf/generic-properties-of-the-eigenvalue-of-the-laplacian-for-1osuaeanhk.pdf

Generic properties of the eigenvalue of the laplacian for compact riemannian manifolds

https://projecteuclid.org/download/pdf_1/euclid.jmsj/1240319489

On the least positive eigenvalue of the Laplacian for compact group manifolds

We study biharmonic maps between Riemannian manifolds with finite energy and finite bi-energy. We show that if the domain is complete and the target of non-positive curvature, then such a map is harmonic. We then give applications to isometric immersions and horizontally conformal submersions.

Biharmonic maps into a Riemannian manifold of non-positive curvature

In this paper, we show that, for every biharmonic submanifold (M, g) of a Riemannian manifold (N, h) with non-positive sectional curvature, if $${\int_M\vert \eta \vert^2 v_g < \infty}$$
 , then (M, g) is minimal in (N, h), i.e., $${\eta\equiv0}$$
 , where η is the mean curvature tensor field of (M, g) in (N, h). This result gives an affirmative answer under the condition $${\int_M\vert \eta \vert^2 v_g < \infty}$$
 to the following generalized Chen’s conjecture: every biharmonic submanifold of a Riemannian manifold with non-positive sectional curvature must be minimal. The conjecture turned out false in case of an incomplete Riemannian manifold (M, g) by a counter example of Ou and Tang (in The generalized Chen’s conjecture on biharmonic sub-manifolds is false, a preprint, 2010).

Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature

In this article, we show that, for a biharmonic hypersurface (M, g) of a Riemannian manifold (N, h) of non-positive Ricci curvature, if ${\int_M\vert H\vert^2 v_g<\infty}$, where H is the mean curvature of (M, g) in (N, h), then (M, g) is minimal in (N, h). Thus, for a counter example (M, g) in the case of hypersurfaces to the generalized Chen’s conjecture (cf. Sect. 1), it holds that ${\int_M\vert H\vert^2 v_g=\infty}$ .

Hajime Urakawa

Papers

Generic properties of the eigenvalue of the laplacian for compact riemannian manifolds

On the least positive eigenvalue of the Laplacian for compact group manifolds

Biharmonic maps into a Riemannian manifold of non-positive curvature

Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature

Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature