# American Institute of Mathematical Sciences

October  1998, 4(4): 709-720. doi: 10.3934/dcds.1998.4.709

## Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric

 1 Dipartimento di Matematica Applicata "U.Dini", Università di Pisa, Via Bonanno 25B - 56126 Pisa, Italy 2 Dip. di Mat. Appl. "U. Dini", Università di Pisa, Italy

Received  October 1996 Revised  April 1998 Published  July 1998

Given a connected compact $C^\infty$ manifold M of dimension $n\ge2$ without boundary, a Riemannian metric $g$ on M and an eigenvalue $\lambda^*(M,g)$ of multiplicity $\nu\ge2$ of the Laplace-Beltrami operator $\Delta_g,$ we provide a sufficient condition such that the set of the deformations of the metric $g,$ which preserve the multiplicity of the eigenvalue, is locally a manifold of codimension $1/2 \nu(\nu+1)-1$ in the space of $C^k$ symmetric covariant 2-tensors on M. Furthermore we prove that such a condition is fulfilled when $n=2$ and $\nu=2.$
Citation: A. M. Micheletti, Angela Pistoia. Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 709-720. doi: 10.3934/dcds.1998.4.709
 [1] Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040 [2] Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078 [3] Andrea Bonito, Roland Glowinski. On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2115-2126. doi: 10.3934/cpaa.2014.13.2115 [4] Qiang Li. A kind of generalized transversality theorem for $C^r$ mapping with parameter. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1043-1050. doi: 10.3934/dcdss.2017055 [5] Mikhail Karpukhin. Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds. Electronic Research Announcements, 2017, 24: 100-109. doi: 10.3934/era.2017.24.011 [6] Peter Giesl. Converse theorem on a global contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5339-5363. doi: 10.3934/dcds.2019218 [7] Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61 [8] Shingo Takeuchi. Partial flat core properties associated to the $p$-laplace operator. Conference Publications, 2007, 2007 (Special) : 965-973. doi: 10.3934/proc.2007.2007.965 [9] Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 311-330. doi: 10.3934/cpaa.2017015 [10] Raffaele Chiappinelli. Eigenvalues of homogeneous gradient mappings in Hilbert space and the Birkoff-Kellogg theorem. Conference Publications, 2007, 2007 (Special) : 260-268. doi: 10.3934/proc.2007.2007.260 [11] J. Tyagi. Multiple solutions for singular N-Laplace equations with a sign changing nonlinearity. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2381-2391. doi: 10.3934/cpaa.2013.12.2381 [12] Antonio Siconolfi, Gabriele Terrone. A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4409-4427. doi: 10.3934/dcds.2012.32.4409 [13] Jochen Brüning, Franz W. Kamber, Ken Richardson. The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations. Electronic Research Announcements, 2010, 17: 138-154. doi: 10.3934/era.2010.17.138 [14] K. Q. Lan. Multiple positive eigenvalues of conjugate boundary value problems with singularities. Conference Publications, 2003, 2003 (Special) : 501-506. doi: 10.3934/proc.2003.2003.501 [15] Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4255-4281. doi: 10.3934/dcds.2021035 [16] Antonella Marini, Thomas H. Otway. Strong solutions to a class of boundary value problems on a mixed Riemannian--Lorentzian metric. Conference Publications, 2015, 2015 (special) : 801-808. doi: 10.3934/proc.2015.0801 [17] Bernard Bonnard, Olivier Cots, Nataliya Shcherbakova. The Serret-Andoyer Riemannian metric and Euler-Poinsot rigid body motion. Mathematical Control & Related Fields, 2013, 3 (3) : 287-302. doi: 10.3934/mcrf.2013.3.287 [18] Roland Gunesch, Philipp Kunde. Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1615-1655. doi: 10.3934/dcds.2018067 [19] Gary Froyland, Simon Lloyd, Anthony Quas. A semi-invertible Oseledets Theorem with applications to transfer operator cocycles. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 3835-3860. doi: 10.3934/dcds.2013.33.3835 [20] George Baravdish, Yuanji Cheng, Olof Svensson, Freddie Åström. Generalizations of $p$-Laplace operator for image enhancement: Part 2. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3477-3500. doi: 10.3934/cpaa.2020152

2020 Impact Factor: 1.392