Exhaustion functions and normal forms for proper maps of balls
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866908649045098496 |
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| author | Lebl, Jiri |
| author_facet | Lebl, Jiri |
| contents | We study a relationship between rational proper maps of balls in different dimensions and strongly plurisubharmonic exhaustion functions of the unit ball induced by such maps. Putting the unique critical point of this exhaustion function at the origin leads to a normal form for rational proper maps of balls. The normal form of the map, which is up to composition with unitaries, takes the origin to the origin, and it normalizes the denominator by eliminating the linear terms and diagonalizing the quadratic part. The singular values of the quadratic part of the denominator are spherical invariants of the map. When these singular values are positive and distinct, the normal form is determined up to a finite subgroup of the unitary group. We also study which denominators arise for cubic maps, and when we do not require taking the origin to the origin, which maps are equivalent to polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_06102 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Exhaustion functions and normal forms for proper maps of balls Lebl, Jiri Complex Variables 32H35, 32U10 We study a relationship between rational proper maps of balls in different dimensions and strongly plurisubharmonic exhaustion functions of the unit ball induced by such maps. Putting the unique critical point of this exhaustion function at the origin leads to a normal form for rational proper maps of balls. The normal form of the map, which is up to composition with unitaries, takes the origin to the origin, and it normalizes the denominator by eliminating the linear terms and diagonalizing the quadratic part. The singular values of the quadratic part of the denominator are spherical invariants of the map. When these singular values are positive and distinct, the normal form is determined up to a finite subgroup of the unitary group. We also study which denominators arise for cubic maps, and when we do not require taking the origin to the origin, which maps are equivalent to polynomials. |
| title | Exhaustion functions and normal forms for proper maps of balls |
| topic | Complex Variables 32H35, 32U10 |
| url | https://arxiv.org/abs/2212.06102 |