Gerhard Huisken

Gerhard Huisken (born 20 May 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.

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 same university 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 1984, he adapted Hamilton's seminal 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.

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 analogy with 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 are elementary via analysis of the Gauss map.

In 1985, he returned to the University of Heidelberg, earning his habilitation in 1986.

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 to constant curvature.

This is measured in terms of the Ricci decomposition.

Almost all of Hamilton's main estimates, particularly the gradient estimate for scalar curvature and the eigenvalue 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 and Richard Schoen established a new convergence theorem for Ricci flow, containing the long-conjectured differentiable sphere theorem as a special case.

Huisken is widely known for his foundational work on the mean curvature flow of hypersurfaces.

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.

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).

Huisken was one of the first authors to consider Richard Hamilton's work on the Ricci flow in higher dimensions.