Tom Ilmanen
Tom Ilmanen | |
|---|---|
| Born | 1961 |
| Died | 2025 (aged 63–64) |
| Education | Ph.D. in Mathematics |
| Alma mater | University of California, Berkeley |
| Occupation | Mathematician |
| Known for | Research in differential geometry, proof of Riemannian Penrose conjecture |
Tom Ilmanen (1961–2025[1]) was an American mathematician specializing in differential geometry and the calculus of variations. He was a professor at ETH Zurich.[2] He obtained his PhD in 1991 at the University of California, Berkeley with Lawrence Craig Evans as supervisor.[3] Ilmanen and Gerhard Huisken used inverse mean curvature flow to prove[4] the Riemannian Penrose conjecture, which is the fifteenth problem in Yau's list of open problems,[5] and was resolved at the same time in greater generality by Hubert Bray using alternative methods.[6]
In their 2001 paper,[4] Huisken and Ilmanen made a conjecture on the mathematics of general relativity, about the curvature in spaces with very little mass: as the mass of the space shrinks to zero, the curvature of the space also shrinks to zero. This was proved in 2023 by Conghan Dong and Antoine Song.[7][8]
In an influential preprint (Singularities of mean curvature flow of surfaces - 1995), Ilmanen conjectured:
For a smooth one-parameter family of closed embedded surfaces in Euclidean 3-space flowing by mean curvature, every tangent flow at the first singular time has multiplicity one. [9]
This has become known as the "multiplicity-one" conjecture. Richard Bamler and Bruce Kleiner proved the multiplicity-one conjecture in a 2023 preprint.[10][11]
Ilmanen received a Sloan Fellowship in 1996.[12]
He wrote the research monograph Elliptic Regularization and Partial Regularity for Motion by Mean Curvature.[13]
Selected publications
[edit]- Huisken, Gerhard, and Tom Ilmanen. "The inverse mean curvature flow and the Riemannian Penrose inequality." Journal of Differential Geometry 59.3 (2001): 353–437. DOI: 10.4310/jdg/1090349447
- Ilmanen, Tom. "Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature." Journal of Differential Geometry 38.2 (1993): 417–461.
- Feldman, Mikhail, Tom Ilmanen, and Dan Knopf. "Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons." Journal of Differential Geometry 65.2 (2003): 169–209.
References
[edit]- ^ "News". Swiss Mathematical Society. Retrieved January 9, 2026.
- ^ "Prof. Dr. Tom Ilmanen". ETH Zurich - Department of Mathematics. 2020-05-11. Retrieved 2025-03-31.
- ^ Tom Ilmanen at the Mathematics Genealogy Project
- ^ a b Huisken, Gerhard; Ilmanen, Tom (2001-11-01). "The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality" (PDF). Journal of Differential Geometry. 59 (3): 353–437. doi:10.4310/jdg/1090349447. ISSN 0022-040X. Retrieved 2025-03-31.
- ^ Greene, Robert Everist; Yau, Shing-Tung (1993). Differential Geometry: Partial Differential Equations on Manifolds. Proceedings of Symposia in Pure Mathematics. Vol. 54.1. doi:10.1090/pspum/054.1. ISBN 978-0-8218-1494-9.
- ^ Mars, Marc (2009). "Present status of the Penrose inequality". Classical and Quantum Gravity. 26 (19). arXiv:0906.5566. doi:10.1088/0264-9381/26/19/193001.
- ^ Nadis, Steve (30 November 2023), "A Century Later, New Math Smooths Out General Relativity", Quanta Magazine
- ^ Dong, Conghan; Song, Antoine (2025). "Stability of Euclidean 3-space for the positive mass theorem". Inventiones Mathematicae. 239 (1): 287–319. arXiv:2302.07414. Bibcode:2025InMat.239..287D. doi:10.1007/s00222-024-01302-z. ISSN 0020-9910.
- ^ Colding, Tobias; Minicozzi, William (2012-03-01). "Generic mean curvature flow I; generic singularities" (PDF). Annals of Mathematics. 175 (2): 755–833. doi:10.4007/annals.2012.175.2.7. ISSN 0003-486X. Retrieved 2025-04-01.
- ^ Nadis, Steve (2025-03-31). "A New Proof Smooths Out the Math of Melting". Quanta Magazine. Retrieved 2025-03-31.
- ^ Bamler, Richard; Kleiner, Bruce (2023). "On the Multiplicity-One Conjecture for Mean Curvature Flows of Surfaces". arXiv:2312.02106 [math.DG].
- ^ "Fellows Database | Alfred P. Sloan Foundation". sloan.org.
- ^ Ilmanen, Tom (1994). Elliptic Regularization and Partial Regularity for Motion by Mean Curvature. Providence, R.I: American Mathematical Soc. ISBN 978-0-8218-2582-2.
