David Allen Hoffman

David Allen Hoffman is an American mathematician whose research concerns

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