Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Plurisubharmonic function
Summary
In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions (which are defined on a Riemannian manifold) plurisubharmonic functions can be defined in full generality on complex analytic spaces.
A function
with domain
is called plurisubharmonic if it is upper semi-continuous, and for every complex line
with
the function is a subharmonic function on the set
In full generality, the notion can be defined on an arbitrary complex manifold or even a complex analytic space as follows. An upper semi-continuous function
is said to be plurisubharmonic if and only if for any holomorphic map
the function
is subharmonic, where denotes the unit disk.
If is of (differentiability) class , then is plurisubharmonic if and only if the hermitian matrix , called Levi matrix, with
entries
is positive semidefinite.
Equivalently, a -function f is plurisubharmonic if and only if is a positive (1,1)-form.
Relation to Kähler manifold: On n-dimensional complex Euclidean space , is plurisubharmonic. In fact, is equal to the standard Kähler form on up to constant multiples. More generally, if satisfies
for some Kähler form , then is plurisubharmonic, which is called Kähler potential. These can be readily generated by applying the ddbar lemma to Kähler forms on a Kähler manifold.
Relation to Dirac Delta: On 1-dimensional complex Euclidean space , is plurisubharmonic. If is a C∞-class function with compact support, then Cauchy integral formula says
which can be modified to
It is nothing but Dirac measure at the origin 0 .
More Examples
If is an analytic function on an open set, then is plurisubharmonic on that open set.
Convex functions are plurisubharmonic
If is a Domain of Holomorphy then is plurisubharmonic
Harmonic functions are not necessarily plurisubharmonic
Plurisubharmonic functions were defined in 1942 by
Kiyoshi Oka and Pierre Lelong.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.