Perspectives and open problems in geometric analysis: spectrum of Laplacian

1. Basic gradient estimate; different variations of the gradient estimates; 2. The theorem of Brascamp-Lieb, Barkey-Émery Riemannian geometry, relation of eigenvalue gap with respect to the first Neumann eigenvalue; the Friedlander-Solomayak theorem, 3. The definition of the Laplacian on L space, theorem of Sturm, 4. Theorem of Wang and its possible generalizations. 1 Gradient estimate of the first eigenvalue Let M be an n-dimensional Riemannian manifold with or without boundary. Let the metric ds be represented by ds = ∑ gijdxidxj , where (x1, · · · , xn) are local coordinates. Let ∆ = 1 √ g ∑ ∂ ∂xi (g √ g ∂ ∂xj ) be the Laplace operator, where (g) = g−1 ij , g = det(gij). The operator ∆ acts on smooth functions. If ∂M 6= ∅, then we may define one of the following two boundary conditions: ¬. Dirichlet condition: f |∂M = 0. ­. Neumann condition: ∂f ∂n |∂M = 0, where n is the outward normal vector of the manifold ∂M .