Spectrum Estimates and Applications to Geometry

In 1867, E. Beltrami (Ann Mat Pura Appl 1(2):329–366, 1867, [12]) introduced a second order elliptic operator on Riemannian manifolds, extending the Laplace operator on Rn, called the Laplace–Beltrami operator. The Laplace–Beltrami operator became one of the most important operators in Mathematics and Physics, playing a fundamental role in differential geometry, geometric analysis, partial differential equations, probability, potential theory, stochastic process, just to mention a few. It is in important in various differential equations that describe physical phenomena such as the diffusion equation for the heat and fluid flow, wave propagation, Laplace equation and minimal surfaces.