Biography:Tom Ilmanen

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Tom Ilmanen
Born1961
Died2025
EducationPh.D. in Mathematics
Alma materUniversity of California, Berkeley
OccupationMathematician
Known forResearch 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 1995 preprint,[9] Ilmanen made the following conjecture:

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. [10]

This has become known as the "multiplicity-one" conjecture. Richard Bamler and Bruce Kleiner proved the multiplicity-one conjecture in a 2023 preprint.[11][12]

Ilmanen received a Sloan Fellowship in 1996.[13]

He wrote the research monograph Elliptic Regularization and Partial Regularity for Motion by Mean Curvature.[14]

Selected publications

  • 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

  1. ↑ "News". Swiss Mathematical Society. https://www.math.ch/. 
  2. ↑ "Prof. Dr. Tom Ilmanen". 2020-05-11. https://math.ethz.ch/research/geometric-analysis-pde/tom-ilmanen.html. 
  3. ↑ Tom Ilmanen at the Mathematics Genealogy Project
  4. ↑ 4.0 4.1 Huisken, Gerhard; Ilmanen, Tom (2001-11-01). "The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality". Journal of Differential Geometry 59 (3): 353–437. doi:10.4310/jdg/1090349447. ISSN 0022-040X. https://projecteuclid.org/journals/journal-of-differential-geometry/volume-59/issue-3/The-Inverse-Mean-Curvature-Flow-and-the-Riemannian-Penrose-Inequality/10.4310/jdg/1090349447.pdf. Retrieved 2025-03-31. 
  5. ↑ Greene, Robert Everist; Yau, Shing-Tung (1993). Differential Geometry: Partial Differential Equations on Manifolds. Proceedings of Symposia in Pure Mathematics. 54.1. doi:10.1090/pspum/054.1. ISBN 978-0-8218-1494-9. 
  6. ↑ Mars, Marc (2009). "Present status of the Penrose inequality". Classical and Quantum Gravity 26 (19). doi:10.1088/0264-9381/26/19/193001. 
  7. ↑ Nadis, Steve (30 November 2023), "A Century Later, New Math Smooths Out General Relativity", Quanta Magazine, https://www.quantamagazine.org/a-century-later-new-math-smooths-out-general-relativity-20231130/ 
  8. ↑ Dong, Conghan; Song, Antoine (2025). "Stability of Euclidean 3-space for the positive mass theorem". Inventiones Mathematicae 239 (1): 287–319. doi:10.1007/s00222-024-01302-z. ISSN 0020-9910. Bibcode: 2025InMat.239..287D. 
  9. ↑ Ilmanen, Tom (1995). "Singularities of mean curvature flow of surfaces". arXiv:2601.21133 [math.DG].
  10. ↑ Colding, Tobias; Minicozzi, William (2012-03-01). "Generic mean curvature flow I; generic singularities". Annals of Mathematics 175 (2): 755–833. doi:10.4007/annals.2012.175.2.7. ISSN 0003-486X. http://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p07-p.pdf. Retrieved 2025-04-01. 
  11. ↑ Nadis, Steve (2025-03-31). "A New Proof Smooths Out the Math of Melting". https://www.quantamagazine.org/a-new-proof-smooths-out-the-math-of-melting-20250331/. 
  12. ↑ Bamler, Richard; Kleiner, Bruce (2023). "On the Multiplicity-One Conjecture for Mean Curvature Flows of Surfaces". arXiv:2312.02106 [math.DG].
  13. ↑ "Fellows Database | Alfred P. Sloan Foundation". https://sloan.org/fellows-database. 
  14. ↑ 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. 



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