120 research outputs found

    Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces

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    Let (M,g)(M,g) be a connected, closed, orientable Riemannian surface and denote by λk(M,g)\lambda_k(M,g) the kk-th eigenvalue of the Laplace-Beltrami operator on (M,g)(M,g). In this paper, we consider the mapping (M,g)↦λk(M,g)(M, g)\mapsto \lambda_k(M,g). We propose a computational method for finding the conformal spectrum Λkc(M,[g0])\Lambda^c_k(M,[g_0]), which is defined by the eigenvalue optimization problem of maximizing λk(M,g)\lambda_k(M,g) for kk fixed as gg varies within a conformal class [g0][g_0] of fixed volume textrmvol(M,g)=1textrm{vol}(M,g) = 1. We also propose a computational method for the problem where MM is additionally allowed to vary over surfaces with fixed genus, γ\gamma. This is known as the topological spectrum for genus γ\gamma and denoted by Λkt(γ)\Lambda^t_k(\gamma). Our computations support a conjecture of N. Nadirashvili (2002) that Λkt(0)=8πk\Lambda^t_k(0) = 8 \pi k, attained by a sequence of surfaces degenerating to a union of kk identical round spheres. Furthermore, based on our computations, we conjecture that Λkt(1)=8π23+8π(k−1)\Lambda^t_k(1) = \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−1k-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 kk-th Laplace-Beltrami eigenvalue has a local maximum with value λk=4π2⌈k2⌉2(⌈k2⌉2−14)−12\lambda_k = 4\pi^2 \left\lceil \frac{k}{2} \right\rceil^2 \left( \left\lceil \frac{k}{2} \right\rceil^2 - \frac{1}{4}\right)^{-\frac{1}{2}}. Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.Comment: 43 pages, 18 figure
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