Schoen assumption

The Schoen conjecture is a problem from the theory of harmonic mappings in the mathematical sub-area of differential geometry . It says that quasi-conformal images between spheres can be continued to quasi-isometric harmonic images of hyperbolic spaces . For 3-dimensional hyperbolic space, it was proven in 2013 by Vladimir Markovic .

Formulation of the Schoen conjecture

Let it be the -dimensional hyperbolic space and its edge in infinity . It is known that every quasi-isometry can be continued to form a quasi-conformal mapping . ${\displaystyle \mathbb {H} ^{n}}$ ${\displaystyle n}$ ${\displaystyle S^{n-1}=\partial _{\infty }\mathbb {H} ^{n}}$ ${\displaystyle F\colon \mathbb {H} ^{n}\to \mathbb {H} ^{n}}$ ${\displaystyle \partial _{\infty }F\colon S^{n-1}\to S^{n-1}}$

The Schoen conjecture for says: For every quasi-conformal mapping ${\displaystyle n\geq 3}$

${\displaystyle f\colon S^{n-1}\to S^{n-1}}$

there is a clear quasi-isometric and harmonic mapping

${\displaystyle F\colon \mathbb {H} ^{n}\to \mathbb {H} ^{n}}$

With

${\displaystyle \partial _{\infty }F=f}$.

For the Schoen conjecture says that there must be quasi-symmetry ${\displaystyle n=2}$

${\displaystyle f\colon S^{1}\to S^{1}}$

a clear harmonic quasi-conformal homeomorphism

${\displaystyle F\colon \mathbb {H} ^{2}\to \mathbb {H} ^{2}}$

with there. (The case was the conjecture originally made by Schoen, the generalization for was made by Li and Wang a little later.) ${\displaystyle \partial _{\infty }F=f}$ ${\displaystyle n=2}$ ${\displaystyle n\geq 3}$

history

The Dirichlet problem for hyperbolic space was solved by Anderson and Sullivan in the 1980s : any continuous function can be continued into a harmonic function . ${\displaystyle g\colon S^{n-1}\to \mathbb {R} }$ ${\displaystyle G\colon \mathbb {H} ^{n}\to \mathbb {R} }$

The Schoen conjecture was established for von Schoen and for von Li and Wang. Li and Wang also proved that the harmonic mapping , if it exists, must be unique. ${\displaystyle n=3}$ ${\displaystyle n>3}$ ${\displaystyle F}$

For - diffeomorphisms (which are then automatically quasi-conform) the Schoen conjecture of Li and Tam was proven. ${\displaystyle C^{1}}$ ${\displaystyle f\colon S^{n-1}\to S^{n-1}}$

The Schoen conjecture was proven by Markovic in 2013 for and 2017 for . ${\displaystyle n=3}$ ${\displaystyle n=2}$

literature

• Peter Li, Jiaping Wang: Harmonic rough isometries into Hadamard space. Asian J. Math. 2 (1998) no. 3, 419-442. online (pdf)
• Peter Li, Luen-Fai Tam: Uniqueness and regularity of proper harmonic maps. Ann. of Math. (2) 137 (1993) no. 1, 167-201. online (pdf)
• Vladimir Markovic: Harmonic maps between 3-dimensional hyperbolic spaces. Invent. Math. 199 (2015), no.3, 921-951. doi : 10.1007 / s00222-014-0536-x
• Vladimir Markovic: J. Amer. Math. Soc. 30 (2017), 799-817. online (pdf)

Individual evidence

1. ↑ Schoen: The role of harmonic mappings in rigidity and deformation problems. Complex geometry (Osaka, 1990), 179-200, Lecture Notes in Pure and Appl. Math., 143, Dekker, New York, 1993.
2. ↑ Li, Wang, op.cit.
3. ↑ Li, Tam, op.cit.
4. ↑ Markovic (2015), op.cit.
5. ↑ Markovic (2017), op.cit.