Heinz Huber

Heinz Huber (born April 25, 1926 in Zofingen ; † December 25, 2000 in Arlesheim ) was a Swiss mathematician who dealt with differential geometry and global analysis. With works that go back to the 1950s, he is one of the founders of the spectral theory of Riemann surfaces .

Life

Heinz Huber came from a humble background. After an internship at Brown, Boveri & Cie. in Baden they noticed him there and made it possible for him to study at the ETH Zurich , where he almost failed the entrance examination because he had insufficient knowledge of Swiss history. He received his doctorate in 1953 under Walter Saxer (and Heinz Hopf ) at the ETH Zurich. His dissertation on analytical mappings of Riemann surfaces in themselves deals with a generalization and geometric interpretation of Émile Picard's Great Theorem . From 1955 he was a professor at the University of Basel .

research

In 1959 he introduced the length spectrum of compact Riemann surfaces, the list of lengths of all closed geodetic tables . He proved a theorem about the asymptotic behavior of this length spectrum, which forms an analogue to the prime number theorem in number theory : The asymptotic (for lengths towards infinity) number of closed geodesics on compact Riemann surfaces with gender greater than 1 and lengths less than or equal is then according to Huber: ${\displaystyle L}$ ${\displaystyle N(L)}$ ${\displaystyle g}$ ${\displaystyle L}$

${\displaystyle N(L)\sim {\frac {e^{L}}{L}}}$

The analogy to the prime number theorem (asymptotic number of prime numbers less than or equal ) results when replacing with . ${\displaystyle x}$ ${\displaystyle L}$ ${\displaystyle \ln(x)}$

In addition, he proved in this work that two compact Riemann surfaces with gender greater than 1 have the same length spectrum if and only if the Laplace operator has the same spectrum of eigenvalues on these Riemann surfaces . The geometric equivalence of length and eigenvalue spectrum and the prime number theorem for geodesic is also ascribed to Atle Selberg (based on Selberg's trace formula from 1956).

His doctoral students include Peter Buser and Christian Blatter .

Fonts

• On analytical mappings of Riemann surfaces in themselves , Comm. Math. Helveticae, Volume 27, 1953, pp. 1–73 (dissertation)
• About a new class of automorphic functions and a lattice point problem in the hyperbolic plane , Part 1 , Comm. Math. Helv., Volume 30, 1956, pp. 20-62
• On the analytical theory of hyperbolic spatial forms and groups of movements , Part 1 , Mathematische Annalen, Volume 138, 1959, pp. 1-26, Part 2 , Math. Annalen, Volume 142, 1960/61, pp. 385-398, supplement , Math. Annalen , Vol. 143, 1961, pp. 463-464
• Lower bounds for the first eigenvalue of the Laplace operator on Riemann surfaces , Comm. Math. Helv., Vol. 61, 1986, pp. 46-59
• About the first eigenvalue of the Laplace operator on compact Riemann surfaces , Comm.Math.Helv., Volume 49, 1974, pp. 251-259
• About the first eigenvalue of the Laplace operator on surfaces of gender two , J. Reine Angewandte Math., Volume 408, 1990, pp. 202-218
• About the eigenvalues ​​of the Laplace operator on compact Riemann surfaces , Comm. Math. Helv., Vol. 51, 1976, pp. 215-231, Part 2 , Comm. Math. Helv., Vol. 53, 1978, pp. 458-469
• About the representations of the automorphism group of a Riemann surface in the eigenspaces of the Laplace operator , Comm. Math. Helv., Vol. 52, 1977, pp. 177-184
• About the dimension of the eigenspaces of the Laplace operator on Riemann surfaces , Comm. Math. Helv., Vol. 55, 1980, pp. 390-397
• About the spectrum of the Laplace operator on compact Riemann surfaces , Comm. Math. Helv., Vol. 57, 1982, pp. 627-647
• Riemann surfaces of the hyperbolic type in Euclidean space , Mathematische Annalen, Volume 139, 1959, pp. 140-146
• About the conformal type of surfaces in Euclidean space , Mathematische Annalen, Volume 146, 1962, pp. 180–188
• Circular disks on Riemann surfaces and eigenvalues ​​of the Laplace operator , J. Reine Angewandte Math., Volume 434, 1993, pp. 191-204
• About grids in the hyperbolic plane , J. Reine Angewandte Math., Volume 458, 1995, pp. 127-156

literature

• Peter Buser Heinz Huber and the length spectrum , in Bruno Colbois, Viktor Schroeder, Christine Riedtmann (editor) math.ch/100, Swiss Mathematical Society 1910–2010 , European Mathematical Society, 2010, p. 163

Individual evidence

1. ↑ according to the curriculum vitae in his dissertation: About analytical maps of Riemann surfaces in themselves doi : 10.3929 / ethz-a-000092402
2. ↑ according to the Chronicle of the Canton of Basel-Landschaft, December 2000
3. ↑ Christian Blatter A Mathematics Study in the 1950s , in Bruno Colbois, Viktor Schroeder, Christine Riedtmann (editor) math.ch/100, Swiss Mathematical Society 1910-2010 , European Mathematical Society, 2010, pdf
4. ^ Mathematics Genealogy Project
5. ↑ On the analytical theory of hyperbolic spatial forms and groups of movements, Part 1