Critical points of Laplace eigenfunctions on polygons

Eigenfunctions of the Laplace operator

The study of the Laplace operator and its corresponding eigenvalue problem is crucial to understand the foundations of 3D shape analysis. For that reason the most important mathematical properties of the Laplace operator in Euclidean spaces, its eigenvalues and eigenfunctions are summarized and explained in this report. The basic definitions and concepts of infinite dimensional function spaces,...

Geodesics 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...

Zeros of real analytic ergodic Laplace eigenfunctions

Heat Kernel Smoothing Using Laplace-Beltrami Eigenfunctions

We present a novel surface smoothing framework using the Laplace-Beltrami eigenfunctions. The Green's function of an isotropic diffusion equation on a manifold is constructed as a linear combination of the Laplace-Beltraimi operator. The Green's function is then used in constructing heat kernel smoothing. Unlike many previous approaches, diffusion is analytically represented as a series expansi...

Quasi-orthogonality on the boundary for Euclidean Laplace eigenfunctions

Consider the Laplacian in a bounded domain in Rd with general (mixed) homogeneous boundary conditions. We prove that its eigenfunctions are ‘quasi-orthogonal’ on the boundary with respect to a certain norm. Boundary orthogonality is proved asymptotically within a narrow eigenvalue window of width o(E1/2) centered about E, as E → ∞. For the special case of Dirichlet boundary conditions, the norm...