Some Examples of ConjugateP-Harmonic Differential Forms

Dirichlet forms: Some infinite dimensional examples

The theory of Dirichlet forms deserves to be better known. It is an area of Markov process theory that uses the energy of functionals to study a Markov process from a quantitative point of view. For instance, the recent notes of Saloff-Coste [S-C] use Dirichlet forms to analyze Markov chains with finite state space, by making energy comparisons. In this way, information about a simple chain is ...

The Harmonic Operator for Exterior Differential Forms.

Differential Operators and Harmonic Weak Maass Forms

For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ2−k (resp. D) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms have transcendental coefficients, we show that those...

Some Caccioppoli Estimates for Differential Forms

Complex manifolds, Kahler metrics, differential and harmonic forms

Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on C rather than C∞ on R. What is the difference betwene holomorphic and C∞? From the PDE point of view, they must satisfy the Cauchy-Riemann equations: f(z) = u+ iv is holomorphic if and only if ∂u ∂x = ∂v ∂y and ∂u ∂y = − ∂v ∂x . The fact that this causes the function f to be analytic(holomorphic)...