David Allen Hoffman is an American mathematician whose research concerns

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

. He is an

adjunct professor An adjunct professor is a type of academic appointment in higher education who does not work at the establishment full-time. The terms of this appointment and the job security of the tenure vary in different parts of the world, however the genera ...

Stanford University Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is consider ...

. In 1985, together with William Meeks, he proved that Costa's surface was embedded. He is a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

since 2018, for "contributions to differential geometry, particularly minimal surface theory, and for pioneering the use of computer graphics as an aid to research." He was awarded the

Chauvenet Prize The Chauvenet Prize is the highest award for mathematical expository writing. It consists of a prize of $1,000 and a certificate, and is awarded yearly by the Mathematical Association of America in recognition of an outstanding expository article ...

in 1990 for his expository article "The Computer-Aided Discovery of New Embedded Minimal Surfaces". He obtained his Ph.D. from

in 1971 under the supervision of

Robert Osserman Robert "Bob" Osserman (December 19, 1926 – November 30, 2011) was an American mathematician who worked in geometry. He is specially remembered for his work on the theory of minimal surfaces. Raised in Bronx, he went to Bronx High School of ...

Technical contributions

In 1973, James Michael and

Leon Simon Leon Melvyn Simon , born in 1945, is a Leroy P. Steele PrizeSee announcemen retrieved 15 September 2017. and Bôcher Memorial Prize, Bôcher Prize-winningSee . mathematician, known for deep contributions to the fields of geometric analysis, g ...

established a

Sobolev inequality In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Re ...

for functions on submanifolds of

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

, in a form which is adapted to the

mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The ...

of the submanifold and takes on a special form for minimal submanifolds. One year later, Hoffman and

Joel Spruck Joel Spruck (born 1946) is a mathematician, J. J. Sylvester Professor of Mathematics at Johns Hopkins University, whose research concerns geometric analysis and elliptic partial differential equations. He obtained his PhD from Stanford University ...

extended Michael and Simon's work to the setting of functions on immersed submanifolds of

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

s. Such inequalities are useful for many problems in

geometric analysis Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of l ...

which deal with some form of prescribed mean curvature. As usual for Sobolev inequalities, Hoffman and Spruck were also able to derive new isoperimetric inequalities for submanifolds of Riemannian manifolds. It is well known that there is a wide variety of

minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...

s in the three-dimensional

. Hoffman and William Meeks proved that any minimal surface which is contained in a half-space must fail to be properly immersed. That is, there must exist a compact set in Euclidean space which contains a noncompact region of the minimal surface. The proof is a simple application of the

maximum principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...

and unique continuation for minimal surfaces, based on comparison with a family of

catenoid In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally describe ...

s. This enhances a result of Meeks,

, and

Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...

, which states that any two complete and properly immersed minimal surfaces in three-dimensional Euclidean space, if both are nonplanar, either have a point of intersection or are separated from each other by a plane. Hoffman and Meeks' result rules out the latter possibility.

Major publications

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References

{{DEFAULTSORT:Hoffman, David Allen 20th-century American mathematicians Fellows of the American Mathematical Society Year of birth missing (living people) Living people 21st-century American mathematicians Stanford University faculty Stanford University alumni