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]
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.
HS74. | Hoffman, David; Spruck, Joel (1974). "Sobolev and isoperimetric inequalities for Riemannian submanifolds". Communications on Pure and Applied Mathematics. 27 (6): 715–727. doi:10.1002/cpa.3160270601. MR 0365424. Zbl 0295.53025. (Erratum: doi:10.1002/cpa.3160280607)
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HM90. | Hoffman, D.; Meeks, W. H. III (1990). "The strong halfspace theorem for minimal surfaces". Inventiones Mathematicae. 101 (2): 373–377. Bibcode:1990InMat.101..373H. doi:10.1007/bf01231506. MR 1062966. S2CID 10695064. Zbl 0722.53054.
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