منابع مشابه
On the nodal sets of toral eigenfunctions
We study the nodal sets of eigenfunctions of the Laplacian on the standard d-dimensional flat torus. The question we address is: Can a fixed hypersurface lie on the nodal sets of eigenfunctions with arbitrarily large eigenvalue? In dimension two, we show that this happens only for segments of closed geodesics. In higher dimensions, certain cylindrical sets do lie on nodal sets corresponding to ...
متن کاملGeodesics and Nodal Sets of Laplace Eigenfunctions on Hyperbolic Manifolds
Let X be a manifold equipped with a complete Riemannian metric of constant negative curvature and finite volume. We demonstrate the finiteness of the collection of totally geodesic immersed hypersurfaces in X that lie in the zero-level set of an arbitrary Laplace eigenfunction. For surfaces, we show that the number can be bounded just in terms of the area of the surface. We also provide constru...
متن کاملA Geometric Covering Lemma and Nodal Sets of Eigenfunctions
The main purpose of this paper is two-fold. On one hand, we prove a sharper covering lemma in Euclidean space Rn for all n ≥ 2 (see Theorem 1.5). On the other hand, we apply this covering lemma to improve existing results for BMO and volume estimates of nodal sets for eigenfunctions u satisfying 4u + λu = 0 on n-dimensional Riemannian manifolds when λ is large (see Theorems 1.7, 1.8). We also i...
متن کاملRestriction of Toral Eigenfunctions to Hypersurfaces and Nodal Sets
We give uniform upper and lower bounds for the L norm of the restriction of eigenfunctions of the Laplacian on the three-dimensional standard flat torus to surfaces with non-vanishing curvature. We also present several related results concerning the nodal sets of eigenfunctions.
متن کاملNodal Sets for Eigenfunctions of the Laplacian on Surfaces
Let M2 be a compact connected surface having a smooth Riemannian metric. The associated Laplacian Ll is a negative definite selfadjoint elliptic operator. Suppose that F is a real eigenfunction, LlF = -)'F. If M has a nonempty boundary, we require Dirichlet or Neumann boundary conditions for F. The nodal set N of F is simply the zero set of F . It contains a finite number of singular points S =...
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ژورنال
عنوان ژورنال: Surveys in Differential Geometry
سال: 2013
ISSN: 1052-9233,2164-4713
DOI: 10.4310/sdg.2013.v18.n1.a7