Maximal multiplicity of Laplacian eigenvalues in negatively curved surfaces (Q6614097)

This article deals with upperbounds of the multiplicity of the eigenvalues of \(-\Delta\), the Laplace Beltrami operator on closed connected Riemanian manifolds. Here is the statement of one typical result proved in this paper. Set, \N\[\N\mathcal{T} = \{(a,b,\rho) \in R^3, a \leq b, \rho>0\}.\N\]\NFor any \((a,b,\rho) \in \mathcal{T} \) let \(\mathcal{M}^{(a,b,\rho)}_g\) be the set of closed connected negatively curved surfaces of genus \(g\) with injectivity radius \(\geq \rho\) and Gaussian curvature \(K(x) \) satisfying \(a \leq K(x) \leq b,\) for any \(x\) in \(M.\) Denote by, \N\[\N0= \lambda_1 < \lambda_2 \leq \cdots \to +\infty\N\]\Nthe discrete spectrum of \(-\Delta\).\N\NThen for any \((a,b,\rho) \in \mathcal{T}\) there exists \(C_0>0\) such that for any \(g \geq 2\) and any \(M \in \mathcal{M}^{(a,b,\rho)}_g\) the multiplicity of \(\lambda_2\) is at most \(C_0 \frac{g}{\text{Log}\text{Log}(1+g)}.\)\N\NThe case \((a,b,\rho) = (-1, -1,\rho)\) corresponds to hyperbolic surfaces with constant curvature \(-1\) and injectivity radius \( \geq \rho.\)\N\NThe above result was only known for hyperbolic surfaces with extra conditions on the number of closed geodesics.\N\NAnother result states that for any \( j \in N, j \geq 2\), any \((a,b,\rho) \in \mathcal{T}\), any \(\beta, K>0\) there exist \(C_0>0\) and \(g_0 \in N, g_0 \geq 2\) such that the number of eigenvalues in \([\lambda_j, \big(1+ \frac{K}{(\text{Log}\,g)^\beta}\lambda_j]\) is at most \(C_0\frac{g}{\text{Log}\text{Log} \,g}\) for any \(g \geq g_0\) and any \(M \in \mathcal{M}^{(a,b,\rho)}_g.\)