# David Allen Hoffman

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

 HS74. David Hoffman and Joel Spruck. Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure Appl. Math. 27 (1974), 715–727. doi:10.1002/cpa.3160270601
 HM90. D. Hoffman and W.H. Meeks III. The strong halfspace theorem for minimal surfaces.   Invent. Math. 101 (1990), no. 2, 373–377. doi:10.1007/bf01231506

## References

1. ^ "David Hoffman | Mathematics". mathematics.stanford.edu.
2. ^ "Costa Surface". minimal.sitehost.iu.edu.
3. ^ "Fellows of the American Mathematical Society". American Mathematical Society.
4. ^ "Chauvenet Prizes | Mathematical Association of America". www.maa.org.
5. ^
6. ^ J.H. Michael and L.M. Simon. Sobolev and mean-value inequalities on generalized submanifolds of n. Comm. Pure Appl. Math. 26 (1973), 361–379.
7. ^ Gerhard Huisken. Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84 (1986), no. 3, 463–480.
8. ^ Richard Schoen and Shing Tung Yau. Proof of the positive mass theorem. II. Comm. Math. Phys. 79 (1981), no. 2, 231–260.
9. ^ William Meeks III, Leon Simon, and Shing Tung Yau. Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature. Ann. of Math. (2) 116 (1982), no. 3, 621–659.