Gerhard Huisken

Gerhard Huisken (born May 20, 1958) is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huisken's monotonicity formula, which is named after him. With Tom Ilmanen, he proved a version of the Riemannian Penrose inequality, which is a special case of the more general Penrose conjecture in general relativity.

Gerhard Huisken
Gerhard Huisken in 2017
Born (1958-05-20) May 20, 1958
Hamburg, Germany
NationalityGerman
Alma materHeidelberg University
Scientific career
FieldsMathematics
InstitutionsUniversity of Tübingen
Doctoral advisorClaus Gerhardt
Doctoral studentsBen Andrews
Simon Brendle

Life

After finishing high school in 1977, Huisken took up studies in mathematics at Heidelberg University. In 1982, one year after his diploma graduation, he completed his PhD at the University of Heidelberg, under the direction of Claus Gerhardt. The topic of his dissertation were non-linear partial differential equations (Reguläre Kapillarflächen in negativen Gravitationsfeldern).

From 1983 to 1984, Huisken was a researcher at the Centre for Mathematical Analysis at the Australian National University (ANU) in Canberra. There, he turned to differential geometry, in particular problems of mean curvature flows and applications in general relativity. In 1985, he returned to the University of Heidelberg, earning his habilitation in 1986. After some time as a visiting professor at the University of California, San Diego, he returned to ANU from 1986 to 1992, first as a Lecturer, then as a Reader. In 1991, he was a visiting professor at Stanford University. From 1992 to 2002, Huisken was a full professor at the University of Tübingen, serving as dean of the faculty of mathematics from 1996 to 1998. From 1999 to 2000, he was a visiting professor at Princeton University.

In 2002, Huisken became a director at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) in Potsdam and, at the same time, an honorary professor at the Free University of Berlin. In April 2013, he took up the post of director at the Mathematical Research Institute of Oberwolfach, together with a professorship at Tübingen University. He remains an external scientific member of the Max Planck Institute for Gravitational Physics.

Huisken's PhD students include Ben Andrews and Simon Brendle, among over twenty-five others.

Work

Huisken's work deals with partial differential equations, differential geometry, and their applications in physics. Numerous phenomena in mathematical physics and geometry are related to surfaces and submanifolds. A dominant theme of Huisken's work has been the study of the deformation of such surfaces, in situations where the rules of deformation are determined by the geometry of those surfaces themselves. Such processes are governed by partial differential equations.

Huisken's contributions to mean curvature flow are particularly fundamental. Through his work, the mean curvature flow of hypersurfaces in various convex settings is largely understood. His discovery of Huisken's monotonicity formula, valid for general mean curvature flows, is a particularly important tool.

In the mathematical study of general relativity, Huisken and Tom Ilmanen (ETH Zurich) were able to prove a significant special case of the Riemannian Penrose inequality. Their method of proof also made a decisive contribution to the inverse mean curvature flow. Hubert Bray later proved a more general version of their result with alternative methods. The general version of the conjecture, which is about black holes or apparent horizons in Lorentzian geometry, is still an open problem (as of 2020).

Mean curvature flow

Huisken is widely known for his foundational work on the mean curvature flow of hypersurfaces. In 1984, he adapted Richard Hamilton's work on the Ricci flow to the setting of mean curvature flow, proving that a normalization of the flow which preserves surface area will deform any smooth closed convex hypersurface of Euclidean space into a round sphere.[1][H84] The major difference between his work and Hamilton's is that, unlike in Hamilton's work, the relevant equation in the proof of the "pinching estimate" is not amenable to the maximum principle. Instead, Huisken made use of iterative integral methods, following earlier work of the analysts Ennio De Giorgi and Guido Stampacchia. In 1987, Huisken adapted his methods to consider an alternative "mean curvature"-driven flow for closed hypersurfaces in Euclidean space, in which the volume enclosed by the surface is kept constant; the result is directly analogous.[H87] Similarly to Hamilton's result, Huisken's results can be viewed as providing proofs that any smooth closed convex hypersurface of Euclidean space is diffeomorphic to a sphere, and is the boundary of a region which is diffeomorphic to a ball. However, both of these results can be proved by more elementary means using the Gauss map.

In 1986, Huisken extended the calculations in his proof to consider hypersurfaces in general Riemannian manifolds.[H86] His result says that if the hypersurface is sufficiently convex relative to the geometry of the Riemannian manifold, then the mean curvature flow will contract it to a point, and that a normalization of surface area in geodesic normal coordinates will give a smooth deformation to a sphere in Euclidean space (as represented by the coordinates). This shows that such hypersurfaces are diffeomorphic to the sphere, and that they are the boundary of a region in the Riemannian manifold which is diffeomorphic to a ball. In this generality, there is not a simple proof using the Gauss map.

Following work of Yoshikazu Giga and Robert Kohn which made extensive use of the Dirichlet energy as weighted by exponentials, Huisken proved in 1990 an integral identity, known as Huisken's monotonicity formula, which shows that, under the mean curvature flow, the integral of the "backwards" Euclidean heat kernel over the evolving hypersurface is always nonincreasing.[2][3][H90] He later extended his formula to allow for general codimension and general positive solutions of the "backwards" heat equation; the monotonicity in this generality crucially uses Richard Hamilton's matrix Li-Yau estimate.[H93][4] An extension to the Riemannian setting was also given by Hamilton.[5] Huisken and Hamilton's ideas were later adapted by Grigori Perelman to the setting of the "backwards" heat equation for volume forms along the Ricci flow.[6]

Huisken and Klaus Ecker made repeated use of the monotonicity result to show that, for a certain class of noncompact graphical hypersurfaces in Euclidean space, the mean curvature flow exists for all positive time and deforms any surface in the class to a self-expanding solution of the mean curvature flow.[EH89] Such a solution moves only by constant rescalings of a single hypersurface. Making use of maximum principle techniques, they were also able to obtain purely local derivative estimates, roughly paralleling those earlier obtained by Wan-Xiong Shi for Ricci flow.[7][EH91]

Given a finite-time singularity of the mean curvature flow, there are several ways to perform microscopic rescalings to analyze the local geometry in regions near points of large curvature. Based on his monotonicity formula, Huisken showed that many of these regions, specifically those known as type I singularities, are modeled in a precise way by self-shrinking solutions of the mean curvature flow.[H90]

There is now a reasonably complete understanding of the rescaling process in the setting of mean curvature flows which only involve hypersurfaces whose mean curvature is strictly positive. Following provisional work by Huisken, Tobias Colding and William Minicozzi have shown that (with some technical conditions) the only self-shrinking solutions of mean curvature flow which have nonnegative mean curvature are the round cylinders, hence giving a complete local picture of the type I singularities in the "mean-convex" setting.[H90][H93][8] In the case of other singular regions, known as type II singularities, Richard Hamilton developed rescaling methods in the setting of Ricci flow which can be transplanted to the mean curvature flow.[9] By modifying the integral methods he developed in 1984, Huisken and Carlo Sinestrari carried out an elaborate inductive argument on the elementary symmetric polynomials of the second fundamental form to show that any singularity model resulting from such rescalings must be a mean curvature flow which moves by translating a single convex hypersurface in some direction.[HSS99a][HS99b] This passage from mean-convexity to full convexity is comparable with the much easier Hamilton-Ivey estimate for Ricci flow, which says that any singularity model of a Ricci flow on a closed 3-manifold must have nonnegative sectional curvature.

Inverse mean curvature flow

In the 1970s, the physicists Robert Geroch, Pong-Soo Jang, and Robert Wald developed ideas connecting the asymptotic behavior of inverse mean curvature flow to the validity of the Penrose conjecture, which relates the energy of an asymptotically flat spacetime to the size of the black holes it contains.[10][11] This can be viewed as a sharpening or quantification of the positive energy theorem, which provides the weaker statement that the energy is nonnegative.

In the 1990s, Yun Gang Chen, Yoshikazu Giga, and Shun'ichi Goto, and independently Lawrence Evans and Joel Spruck, developed a theory of weak solutions for mean curvature flow by considering level sets of solutions of a certain elliptic partial differential equation.[12][13] Tom Ilmanen made progress on understanding the theory of such elliptic equations, via approximations by elliptic equations of a more standard character.[14] Huisken and Ilmanen were able to adapt these methods to the inverse mean curvature flow, thereby making the methodology of Geroch, Jang, and Wald mathematically precise. Their result deals with noncompact three-dimensional Riemannian manifolds-with-boundary of nonnegative scalar curvature whose boundary is minimal, relating the geometry near infinity to the surface area of the largest boundary component.[HI01] Hubert Bray, by making use of the positive mass theorem instead of the inverse mean curvature flow, was able to improve Huisken and Ilmanen's inequality to involve the total surface area of the boundary.[15]

Honours and awards

Huisken is a fellow of the Heidelberg Academy for Sciences and Humanities, the Berlin-Brandenburg Academy of Sciences and Humanities, the Academy of Sciences Leopoldina, and the American Mathematical Society.[16]

Major publications
H84.Gerhard Huisken. Flow by mean curvature of convex surfaces into spheres. J. Differential Geom. 20 (1984), no. 1, 237–266. doi:10.4310/jdg/1214438998
H86.Gerhard Huisken. Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84 (1986), no. 3, 463–480. doi:10.1007/BF01388742
H87.Gerhard Huisken. The volume preserving mean curvature flow. J. Reine Angew. Math. 382 (1987), 35–48. doi:10.1515/crll.1987.382.35
EH89.Klaus Ecker and Gerhard Huisken. Mean curvature evolution of entire graphs. Ann. of Math. (2) 130 (1989), no. 3, 453–471. doi:10.2307/1971452
H90.Gerhard Huisken. Asymptotic behavior for singularities of the mean curvature flow. J. Differential Geom. 31 (1990), no. 1, 285–299. doi:10.4310/jdg/1214444099
EH91.Klaus Ecker and Gerhard Huisken. Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105 (1991), no. 3, 547–569. doi:10.1007/BF01232278
H93.Gerhard Huisken. Local and global behaviour of hypersurfaces moving by mean curvature. Proc. Sympos. Pure Math., 54, Part 1 (1993), pp. 175–191. Differential Geometry: Partial Differential Equations on Manifolds (Proceedings of the AMS Summer Research Institute on Differential Geometry held at the University of California, Los Angeles, California, July 8–28, 1990). Amer. Math. Soc., Providence, RI. Edited by Robert Greene and S.T. Yau. doi:10.1090/pspum/054.1
HS99a.Gerhard Huisken and Carlo Sinestrari. Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differential Equations 8 (1999), no. 1, 1–14. doi:10.1007/s005260050113
HS99b.Gerhard Huisken and Carlo Sinestrari. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183 (1999), no. 1, 45–70. doi:10.1007/BF02392946
HI01.Gerhard Huisken and Tom Ilmanen. The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differential Geom. 59 (2001), no. 3, 353–437. doi:10.4310/jdg/1090349447

References
  1. Richard S. Hamilton. Three-manifolds with positive Ricci curvature. J. Differential Geometry 17 (1982), no. 2, 255–306.
  2. Yoshikazu Giga and Robert V. Kohn. Asymptotically self-similar blow-up of semilinear heat equations. Comm. Pure Appl. Math. 38 (1985), no. 3, 297–319.
  3. Yoshikazu Giga and Robert V. Kohn. Characterizing blowup using similarity variables. Indiana Univ. Math. J. 36 (1987), no. 1, 1–40.
  4. Richard S. Hamilton. A matrix Harnack estimate for the heat equation. Comm. Anal. Geom. 1 (1993), no. 1, 113–126.
  5. Richard S. Hamilton. Monotonicity formulas for parabolic flows on manifolds. Comm. Anal. Geom. 1 (1993), no. 1, 127–137.
  6. Grisha Perelman. The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159
  7. Wan-Xiong Shi. Deforming the metric on complete Riemannian manifolds. J. Differential Geom. 30 (1989), no. 1, 223–301.
  8. Tobias H. Colding and William P. Minicozzi, II. Generic mean curvature flow I: generic singularities. Ann. of Math. (2) 175 (2012), no. 2, 755–833.
  9. Richard S. Hamilton. The formation of singularities in the Ricci flow. Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7–136. Int. Press, Cambridge, MA, 1995.
  10. Robert Geroch. Energy extraction. Ann. New York Acad. Sci. 224 (1973), 108–117.
  11. Pong Soo Jang and Robert M. Wald. The positive energy conjecture and the cosmic censor hypothesis. J. Mathematical Phys. 18 (1977), no. 1, 41–44.
  12. Yun Gang Chen, Yoshikazu Giga, and Shun'ichi Goto. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geom. 33 (1991), no. 3, 749–786.
  13. L.C. Evans and J. Spruck. Motion of level sets by mean curvature. I. J. Differential Geom. 33 (1991), no. 3, 635–681.
  14. Tom Ilmanen. Elliptic regularization and partial regularity for motion by mean curvature. Mem. Amer. Math. Soc. 108 (1994), no. 520, x+90 pp.
  15. Hubert L. Bray. Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differential Geom. 59 (2001), no. 2, 177–267.
  16. List of Fellows of the American Mathematical Society, retrieved 2013-07-07.
  17. Huisken, Gerhard (1998). "Evolution of hypersurfaces by their curvature in Riemannian manifolds". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. pp. 349–360.

Media related to Gerhard Huisken (mathematician) at Wikimedia Commons

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.