---
tag: MT9P2X85V8
kind: concept
name: Elliptic operator on a compact manifold
msc: [58J05, 47F10, 35J46]
sources:
  - url: https://en.wikipedia.org/wiki/Elliptic_operator
---

## Summary

An elliptic operator on a compact manifold is a [[differential operator between vector bundles|CGBX1N9V8H]] whose [[principal symbol|S8GC34D5ZG]] is invertible at every nonzero covector.
It extends the notion of an [[elliptic operator|XY8GEQSWPR]] on a domain of $\R^n$ to sections of bundles on a closed manifold.
On a compact manifold such an operator is Fredholm between suitable Sobolev spaces, so it has a finite index.
The Hodge Laplacian, the operator $d + d^*$ on forms, Dirac operators and the operator $\bar\partial + \bar\partial^*$ built from the Dolbeault operator are the main examples.
The Atiyah–Singer index theorem computes their indices.

## Explanations

### L1 human

Ellipticity says that the operator controls every derivative of its input, in every direction, at the top order.
On a compact manifold there is no boundary and no infinity where things can escape.
Together these two facts make an elliptic operator nearly invertible: only finitely many independent inputs are killed and only finitely many independent outputs are missed.
For example, the Laplacian on the circle kills only the constant functions, and its image consists of the functions with average zero.