David Allen Hoffman is an American mathematician whose research concerns differential geometry. He is an adjunct professor at Stanford University. ^[1] In 1985, together with William Meeks, he proved that Costa's surface was embedded. ^[2] He is a fellow of the American Mathematical Society since 2018, for "contributions to differential geometry, particularly minimal surface theory, and for pioneering the use of computer graphics as an aid to research." ^[3] He was awarded the Chauvenet Prize in 1990 for his expository article "The Computer-Aided Discovery of New Embedded Minimal Surfaces". ^[4] He obtained his Ph.D. from Stanford University in 1971 under the supervision of Robert Osserman. ^[5]

Technical contributions

In 1973, James Michael and Leon Simon established a Sobolev inequality for functions on submanifolds of Euclidean space, in a form which is adapted to the mean curvature of the submanifold and takes on a special form for minimal submanifolds. ^[6] One year later, Hoffman and Joel Spruck extended Michael and Simon's work to the setting of functions on immersed submanifolds of Riemannian manifolds. ^[HS74] Such inequalities are useful for many problems in geometric analysis which deal with some form of prescribed mean curvature. ^{[7] [8]} As usual for Sobolev inequalities, Hoffman and Spruck were also able to derive new isoperimetric inequalities for submanifolds of Riemannian manifolds. ^[HS74]

It is well known that there is a wide variety of minimal surfaces in the three-dimensional Euclidean space. Hoffman and William Meeks proved that any minimal surface which is contained in a half-space must fail to be properly immersed. ^[HM90] 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 and unique continuation for minimal surfaces, based on comparison with a family of catenoids. This enhances a result of Meeks, Leon Simon, and Shing-Tung Yau, 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. ^[9] Hoffman and Meeks' result rules out the latter possibility.

Major publications

References