Laplacian comparison and sub-mean-value theorem for multiplier Hermitian manifolds

On Multiplier Hermitian Structures on Compact Kähler Manifolds

Mean Value Theorems on Manifolds

We derive several mean value formulae on manifolds, generalizing the classical one for harmonic functions on Euclidean spaces as well as the results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation a mean value theorem with respect to ‘heat spheres’ is proved for heat equation with respect to evolving Riemannian metrics via a space-time consideration. Som...

Mean Value Properties of the Laplacian

Let <¡>(z2) be an even entire function of temperate exponential type, L a selfadjoint realization of -A + c(x), where A is the Laplace-Beltrami operator on a Riemannian manifold, and <¡>(L) the operator given by spectral theory. A PaleyWiener theorem on the support of ( L) is proved, and is used to show that Lu = \u on a suitable domain implies (L)u = <p(\)u, as well as a generalization o...

The First Mean Value Theorem for Integrals

For simplicity, we use the following convention: X is a non empty set, S is a σ-field of subsets of X, M is a σ-measure on S, f , g are partial functions from X to R, and E is an element of S. One can prove the following three propositions: (1) If for every element x of X such that x ∈ dom f holds f(x) ≤ g(x), then g − f is non-negative. (2) For every set Y and for every partial function f from...

The Mean Value Theorem and Its Consequences

The point (M,f(M)) is called an absolute maximum of f if f(x) ≤ f(M) for every x in the domain of f . The point (m, f(m)) is called an absolute minimum of f if f(x) ≥ f(m) for every x in the domain of f . More than one absolute maximum or minimum may exist. For example, if f(x) = |x| for x ∈ [−1, 1] then f(x) ≤ 1 and there are absolute maxima at (1, 1) and at (−1, 1), but only one absolute mini...