Maximization of Laplace−Beltrami eigenvalues on closed Riemannian surfaces

Kao, Chiu-Yen; Lai, Rongjie; Osting, Braxton

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 2, pp. 685-720.

Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by $\lambda {}_k(M,g)$ the $k$th eigenvalue of the Laplace−Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M,g)\to \lambda {}_k(M,g)$. We propose a computational method for finding the conformal spectrum ${\Lambda }_k^c(M,\left[g_0\right])$, which is defined by the eigenvalue optimization problem of maximizing $\lambda {}_k(M,g)$ for $k$ fixed as $g$ varies within a conformal class $\left[g_0\right]$ of fixed volume vol$(M,g)=1$. We also propose a computational method for the problem where $M$ is additionally allowed to vary over surfaces with fixed genus, $\gamma$. This is known as the topological spectrum for genus $\gamma$ and denoted by ${\Lambda }_k^t\left(\gamma \right)$. Our computations support a conjecture of [N. Nadirashvili, J. Differ. Geom. 61 (2002) 335–340.] that ${\Lambda }_k^t\left(0\right)=8\pi k$, attained by a sequence of surfaces degenerating to a union of $k$ identical round spheres. Furthermore, based on our computations, we conjecture that ${\Lambda }_k^t\left(1\right)=\frac{8{\pi }^2}{\sqrt{3}}+8\pi (k-1)$, attained by a sequence of surfaces degenerating into a union of an equilateral flat torus and $k-1$ identical round spheres. The values are compared to several surfaces where the Laplace−Beltrami eigenvalues are well-known, including spheres, flat tori, and embedded tori. In particular, we show that among flat tori of volume one, the $k$th Laplace−Beltrami eigenvalue has a local maximum with value ${\lambda }_k=4{\pi }^2{\lceil \frac{k}{2}\rceil }^2{({\lceil \frac{k}{2}\rceil }^2-\frac{1}{4})}^{-\frac{1}{2}}$. Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.

Classification : 35P15, 49Q10, 65N25, 58J50, 58C40
Mots clés : Extremal Laplace−Beltrami eigenvalues, conformal spectrum, topological spectrum, closed Riemannian surface, spectral geometry, isoperimetric inequality

P.R.S. Antunes and P. Freitas, Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians. J. Optim. Theory Appl. 154 (2012) 235–257. | DOI | MR | Zbl

M.G. Armentano and R.G. Durán, Asymptotic lower bounds for eigenvalues by nonconforming finite element methods. Electron. Trans. Numer. Anal. 17 (2004) 93–101. | MR | Zbl

M.S. Ashbaugh and R.D. Benguria, Isoperimetric inequalities for eigenvalues of the Laplacian. Proc. Symp. Pure Math. 76 (2007) 105–139. | DOI | MR | Zbl

P.H. Bérard and G. Besson, Lectures on Spectral Geometry. Conselho Nacional de Desenvolvimento Cientifico e Tecnologico, Instituto de Matematica Pura e Aplicada (1985). | Zbl

M. Berger, Sur les premiéres valeurs propres des variétés Riemanniennes. Comp. Math. 26 (1973) 129–149. | Numdam | MR | Zbl

D. Boffi, Finite element approximation of eigenvalue problems. Acta Numer. 19 (2010) 1–120. | DOI | MR | Zbl

P. Buser, Geometry and spectra of compact Riemann surfaces. Springer (2010). | MR | Zbl

B. Colbois and J. Dodziuk, Riemannian metrics with large $\lambda {}_1$, Proc. Am. Math. Soc. 122 (1994) 905–906. | MR | Zbl

B. Colbois and A. El Soufi, Extremal eigenvalues of the Laplacian in a conformal class of metrics: the ‘conformal spectrum’. Ann. Global Anal. Geom. 24 (2003) 337–349. | DOI | MR | Zbl

B. Colbois and A. El Soufi, Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces. Math. Z. 278 (2014) 529–546. | DOI | MR | Zbl

B. Colbois, A. El Soufi and A. Girouard, Isoperimetric control of the spectrum of a compact hypersurface. J. Reine Angew. Math. 2013 (2013) 49–65. | DOI | MR | Zbl

B. Colbois, E.B. Dryden and A. El Soufi, Bounding the eigenvalues of the Laplace–Beltrami operator on compact submanifolds. Bull. London Math. Soc. 42 (2010) 96–108. | DOI | MR | Zbl

I. Chavel, Eigenvalues in Riemannian geometry. Academic Press (1984). | MR | Zbl

S. Cox and J. Mclaughlin, Extremal eigenvalue problems for composite membranes. Part I and II. Appl. Math. Optim. 22 (1990) 153–167 and 169–187. | DOI | MR | Zbl

T.A. Driscoll, N. Hale and L.N. Trefethen, Chebfun guide. Pafnuty Publications, Oxford (2014).

A. El Soufi and S. Ilias, Laplacian eigenvalue functionals and metric deformations on compact manifolds. J. Geom. Phys. 58 (2008) 89–104. | DOI | MR | Zbl

S. Friedland, Extremal eigenvalue problems defined on conformal classes of compact Riemannian manifolds. Comment. Math. Helvetici 54 (1979) 494–507. | DOI | MR | Zbl

O. Giraud and K. Thas, Hearing shapes of drums-mathematical and physical aspects of isospectrality. Rev. Mod. Phys. 82 (2010) 2213–2255. | DOI

A. Girouard, N. Nadirashvili and I. Polterovich, Maximization of the second positive Neumann eigenvalue for planar domains. J. Differ. Geom. 83 (2009) 637–661. | DOI | MR | Zbl

R. Glowinski and D.C. Sorensen, Computing the eigenvalues of the Laplace-Beltrami operator on the surface of a torus: A numerical approach. Partial Differential Equations. In vol. 16 of Comput. Methods Appl. Sci. Springer (2008) 225–232. | MR | Zbl

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators. Verlag, Birkhäuser (2006). | MR | Zbl

J. Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogenes. (French). C. R. Acad. Sci. Paris Sér. AB 270 (1970) A1645–A1648. | MR | Zbl

Y. Imayoshi and M. Taniguchi, An Introduction to Teichmüller Spaces. Springer-Verlag (1992). | MR | Zbl

D. Jakobson, M. Levitin, N. Nadirashvili, N. Nigam and I. Polterovich, How large can the first eigenvalue be on a surface of genus two? Int. Math. Res. Not. 2005 (2005) 3967–3985. | DOI | MR | Zbl

D. Jakobson, N. Nadirashvili and I. Polterovich, Extremal metric for the first eigenvalue on a Klein bottle. Canadian J. Math. 58 (2006) 381–400. | DOI | MR | Zbl

M.A. Karpukhin, Nonmaximality of known extremal metrics on torus and Klein bottle. Sb. Math. 204 (2013) 1–17. | DOI | MR | Zbl

M.A. Karpukhin, Spectral properties of bipolar surfaces to Otsuki tori. J. Spectr. Theory 4 (2014) 87–111. | DOI | MR | Zbl

G. Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces. Adv. Math. 258 (2014) 191–239. | DOI | MR | Zbl

N. Korevaar, Upper bounds for eigenvalues of conformal metrics. J. Differ. Geom. 37 (1993) 73–93. | DOI | MR | Zbl

P. Kroger, On the spectral gap for compact manifolds. J. Differ. Geom. 36 (1992), 315–330. | DOI | MR | Zbl

R. Lai, Z. Wen, W. Yin, X. Gu and L.M. Lui, Folding-free global conformal mapping for genus-0 surfaces by harmonic energy minimization, J. Sci. Comput. 18 (2014) 705–725. | DOI | MR | Zbl

H. Lapointe, Spectral properties of bipolar minimal surfaces in $S^4$. Differ. Geom. Appl. 26 (2008) 9–22. | DOI | MR | Zbl

S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods. New York (2005). | MR | Zbl

R.S. Laugesen and B.A. Siudeja, Sums of Laplace eigenvalues: Rotations and tight frames in higher dimensions. J. Math. Phys. 52 (2011) 093703. | DOI | MR | Zbl

R.B. Lehoucq and D.C. Sorensen, Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J. Matrix Anal. Appl. 17 (1996) 789–821. | DOI | MR | Zbl

A.S. Lewis and M.L. Overton, Nonsmooth optimization via quasi-Newton methods. Math. Program. 141 (2013) 135–163. | DOI | MR | Zbl

J. Ling and Z. Lu, Bounds of eigenvalues on Riemannian manifolds. ALM 10 (2010) 241–264. | MR | Zbl

J. Milnor, Eigenvalues of the Laplace operator on certain manifolds. Proc. Nat. Acad. Sci. USA 51 (1964) 542. | DOI | MR | Zbl

N. Nadirashvili, Berger’s isoperimetric problem and minimal immersions of surfaces. Geom. Funct. Anal. 6 (1996) 877–897. | DOI | MR | Zbl

N. Nadirashvili, Isoperimetric inequality for the second eigenvalue of a sphere. J. Differ. Geom. 61 (2002) 335–340. | DOI | MR | Zbl

N. Nadirashvili and Y. Sire, Conformal spectrum and harmonic maps. Preprint (2014). | arXiv | MR

B. Osting, Optimization of spectral functions of Dirichlet–Laplacian eigenvalues. J. Comp. Phys. 229 (2010) 8578–8590. | DOI | MR | Zbl

B. Osting and C.Y. Kao, Minimal convex combinations of sequential Laplace-Dirichlet eigenvalues. SIAM J. Sci. Comput. 35 (2013) B731–B750. | DOI | MR | Zbl

B. Osting and C.Y. Kao, Minimal convex combinations of three sequential Laplace–Dirichlet eigenvalues. Appl. Math. Optim. 69 (2014) 123–139. | DOI | MR | Zbl

E. Oudet, Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM: COCV 10 (2004) 315–330. | Numdam | MR | Zbl

F. Pacard and P. Sicbaldi, Extremal domains for the first eigenvalue of the Laplace-Beltrami operator. Ann. Inst. Fourier 59 (2009) 515–542. | DOI | Numdam | MR | Zbl

A.V. Penskoi, Extremal spectral properties of Lawson tau-surfaces and the Lamé equation. Moscow Math. J. 12 (2012) 173–192. | DOI | MR | Zbl

A.V. Penskoi, Extremal metrics for eigenvalues of the Laplace-Beltrami operator on surfaces. Russian Math. Surveys 68 (2013) 1073. | DOI | MR | Zbl

A.V. Penskoi, Extremal spectral properties of Otsuki tori. Math. Nachr. 286 (2013) 379–391. | DOI | MR | Zbl

A.V. Penskoi, Generalized Lawson tori and Klein bottles. J. Geom. Anal. 25 (2015) 2645–2666. | DOI | MR | Zbl

R. Petrides, Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces. Preprint (2013). | arXiv | MR | Zbl

R. Petrides, Maximization of the second conformal eigenvalue of spheres. Proc. Am. Math. Soc. 142 (2014) 2385–2394. | DOI | MR | Zbl

A. Qiu, D. Bitouk and M.I. Miller, Smooth functional and structural maps on the neocortex via orthonormal bases of the Laplace–Beltrami operator. IEEE Trans. Med. Imaging 25 (2006) 1296–1306. | DOI

J.W.S. Rayleigh, The Theory of Sound. Vol. 1. Dover Publications New York (1877). | Zbl

M. Reuter, F.-E. Wolter and N. Peinecke, Laplace–Beltrami spectra as ‘Shape-DNA’ of surfaces and solids. Comput. Aid. Design 38 (2006) 342–366. | DOI

R. Schoen and S.-T. Yau, Lectures on differential geometry. International Press (1994). | Zbl

Y. Shi, R. Lai, R. Gill, D. Pelletier, D. Mohr, N. Sicotte and A.W. Toga, Conformal metric optimization on surface (cmos) for deformation and mapping in Laplace-Beltrami embedding space, Medical Image Computing and Computer-Assisted Intervention–MICCAI 2011. Springer (2011) 327–334.

D. C. Sorensen, Implicit application of polynomial filters in a $k$-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13 (1992) 357–385. | DOI | Zbl

L. N. Trefethen, Spectral methods in MATLAB. Vol. 10. SIAM (2000). | Zbl

H. Urakawa, On the least positive eigenvalue of the Laplacian for compact group manifolds. J. Math. Soc. Jpn 31 (1979) 209–226. | DOI | Zbl

P.C. Yang and S.-T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds. Ann. Sc. Norm. Super. Pisa-Cl. Sci. 7 (1980) 55–63. | Numdam | Zbl