Tom Ilmanen

Tom Ilmanen
Born1961 (age 63–64)
EducationPh.D. in Mathematics
Alma materUniversity of California, Berkeley
OccupationMathematician
Known forResearch in differential geometry, proof of Riemannian Penrose conjecture

Tom Ilmanen (1961 - 2025) was an American mathematician specializing in differential geometry and the calculus of variations. He was a professor at ETH Zurich.[1] He obtained his PhD in 1991 at the University of California, Berkeley with Lawrence Craig Evans as supervisor.[2] Ilmanen and Gerhard Huisken used inverse mean curvature flow to prove[3] the Riemannian Penrose conjecture, which is the fifteenth problem in Yau's list of open problems,[4] and was resolved at the same time in greater generality by Hubert Bray using alternative methods.[5]

In their 2001 paper,[3] 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.[6][7]

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. [8]

This has become known as the "multiplicity-one" conjecture. Richard Bamler and Bruce Kleiner proved the multiplicity-one conjecture in a 2023 preprint.[9][10]

Ilmanen received a Sloan Fellowship in 1996.[11]

He wrote the research monograph Elliptic Regularization and Partial Regularity for Motion by Mean Curvature.[12]

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. ^ "Prof. Dr. Tom Ilmanen". ETH Zurich - Department of Mathematics. 2020-05-11. Retrieved 2025-03-31.
  2. ^ Tom Ilmanen at the Mathematics Genealogy Project
  3. ^ 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.
  4. ^ 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.
  5. ^ 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.
  6. ^ Nadis, Steve (30 November 2023), "A Century Later, New Math Smooths Out General Relativity", Quanta Magazine
  7. ^ 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.
  8. ^ 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.
  9. ^ Nadis, Steve (2025-03-31). "A New Proof Smooths Out the Math of Melting". Quanta Magazine. Retrieved 2025-03-31.
  10. ^ Bamler, Richard; Kleiner, Bruce (2023). "On the Multiplicity-One Conjecture for Mean Curvature Flows of Surfaces". arXiv:2312.02106 [math.DG].
  11. ^ "Fellows Database | Alfred P. Sloan Foundation". sloan.org.
  12. ^ 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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