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Gerhard Huisken
Gerhard Huisken in 2017
Born	(1958-05-20)20 May 1958 (age 67) Hamburg,Germany
Alma mater	Heidelberg University
Known for	Huisken's monotonicity formula
Scientific career
Fields	Mathematics
Institutions	Australian National University University of Tübingen Max Planck Institute for Gravitational Physics Mathematical Research Institute of Oberwolfach
Thesis	Reguläre Kapillarflächen in negativen Gravitationsfeldern (1983)
Doctoral advisor	Claus Gerhardt
Doctoral students	Ben Andrews Simon Brendle

Gerhard Huisken (born 20 May 1958) is a Germanmathematician whose research concernsdifferential geometry andpartial differential equations. He is known for foundational contributions to the theory of themean curvature flow, includingHuisken's monotonicity formula, which is named after him. WithTom Ilmanen, he proved a version of theRiemannian Penrose inequality, which is a special case of the more general Penrose conjecture ingeneral relativity.

After finishing high school in 1977, Huisken took up studies inmathematics atHeidelberg University. In 1982, one year after his diploma graduation, he completed his PhD at the same university under the direction ofClaus 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 theAustralian National University (ANU) in Canberra. There, he turned todifferential geometry, in particular problems ofmean curvature flows and applications ingeneral relativity. In 1985, he returned to the University of Heidelberg, earning hishabilitation in 1986. After some time as a visiting professor at theUniversity 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 atStanford University. From 1992 to 2002, Huisken was a full professor at theUniversity of Tübingen, serving as dean of the faculty of mathematics from 1996 to 1998. From 1999 to 2000, he was a visiting professor atPrinceton University.

In 2002, Huisken became a director at theMax Planck Institute for Gravitational Physics (Albert Einstein Institute) inPotsdam and, at the same time, an honorary professor at theFree University of Berlin. In April 2013, he took up the post of director at theMathematical 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 includeBen Andrews andSimon Brendle, among over twenty-five others.

Huisken's work deals withpartial differential equations,differential geometry, and their applications inphysics. Numerous phenomena inmathematical physics and geometry are related to surfaces andsubmanifolds. 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 tomean curvature flow are particularly fundamental. Through his work, the mean curvature flow of hypersurfaces in variousconvex settings is largely understood. His discovery ofHuisken's monotonicity formula, valid for general mean curvature flows, is a particularly important tool.

In the mathematical study ofgeneral relativity, Huisken andTom Ilmanen (ETH Zurich) were able to prove a significant special case of theRiemannian Penrose inequality. Their method of proof also made a decisive contribution to theinverse 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 aboutblack holes orapparent horizons inLorentzian geometry, is still anopen problem (as of 2020).

Huisken was one of the first authors to considerRichard Hamilton's work on theRicci flow in higher dimensions.^[1] In 1985, Huisken published a version of Hamilton's analysis in arbitrary dimensions, in which Hamilton's assumption of the positivity of Ricci curvature is replaced by a quantitative closeness toconstant curvature.^[H85] This is measured in terms of theRicci decomposition. Almost all of Hamilton's main estimates, particularly thegradient estimate for scalar curvature and theeigenvalue pinching estimate, were put by Huisken into the context of general dimensions.

Several years later, the validity of Huisken's convergence theorems were extended to broader curvature conditions via new algebraic ideas of Christoph Böhm and Burkhard Wilking. In a major application of Böhm and Wilking's work, Brendle andRichard Schoen established a new convergence theorem for Ricci flow, containing the long-conjectureddifferentiable sphere theorem as a special case.

Huisken is widely known for his foundational work on themean curvature flow ofhypersurfaces. In 1984, he adapted Hamilton's seminal work on theRicci flow to the setting of mean curvature flow, proving that a normalization of the flow which preserves surface area will deform any smooth closedconvex hypersurface ofEuclidean space into a round sphere.^[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 themaximum principle. Instead, Huisken made use of iterative integral methods, following earlier work of the analystsEnnio De Giorgi andGuido Stampacchia. In analogy with Hamilton's result, Huisken's results can be viewed as providing proofs that any smooth closed convex hypersurface of Euclidean space isdiffeomorphic to a sphere, and is the boundary of a region which is diffeomorphic to a ball. However, both of these results are elementary via analysis of theGauss map.

Later, Huisken extended the calculations in his proof to consider hypersurfaces in generalRiemannian 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 ingeodesic 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.

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] Later, in collaboration withShing-Tung Yau, this work was extended to Riemannian settings.^[HY96] The corresponding existence and convergence result of Huisken–Yau illustrates a geometric phenomena of manifolds withpositive ADM mass, namely that they are foliated by surfaces ofconstant mean curvature. With a corresponding uniqueness result, they interpreted this foliation as a measure ofcenter of mass in the theory ofgeneral relativity.

Following work of Yoshikazu Giga andRobert Kohn which made extensive use of theDirichlet energy as weighted by exponentials, Huisken proved in 1990 an integral identity, known asHuisken's monotonicity formula, which shows that, under the mean curvature flow, the integral of the "backwards" Euclideanheat 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 usesRichard 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 byGrigori Perelman to the setting of the "backwards" heat equation forvolume forms along theRicci 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 aself-expanding solution of the mean curvature flow.^[EH89] Such a solution moves only by constant rescalings of a single hypersurface. Making use ofmaximum 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 largecurvature. Based on his monotonicity formula, Huisken showed that many of these regions, specifically those known astype I singularities, are modeled in a precise way byself-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 whosemean curvature is strictly positive. Following provisional work by Huisken,Tobias Colding andWilliam 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 astype 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 elementarysymmetric polynomials of thesecond 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 nonnegativesectional curvature.

In the 1970s, the physicistsRobert Geroch, Pong-Soo Jang, andRobert Wald developed ideas connecting the asymptotic behavior ofinverse mean curvature flow to the validity of the Penrose conjecture, which relates theenergy of an asymptotically flat spacetime to the size of theblack holes it contains.^[10][11] This can be viewed as a sharpening or quantification of thepositive 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 independentlyLawrence Evans andJoel Spruck, developed a theory ofweak solutions for mean curvature flow by consideringlevel sets of solutions of a certainelliptic 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 nonnegativescalar curvature whose boundary isminimal, relating the geometry near infinity to the surface area of the largest boundary component.^[HI01]Hubert Bray, by making use of thepositive 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]

Huisken is a fellow of theHeidelberg Academy for Sciences and Humanities, theBerlin-Brandenburg Academy of Sciences and Humanities, theAcademy of Sciences Leopoldina, and theAmerican Mathematical Society.^[16]

• 1991: Medal of theAustralian Mathematical Society
• 1998:invited speaker at the International Congress of Mathematicians^[17]
• 2002:Gauss Lecture of theGerman Mathematical Society
• 2003:Gottfried Wilhelm Leibniz Prize

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