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]

For other people named David Hoffman, see David Hoffman (disambiguation).
Not to be confused with David A. Huffman.

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.
    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)
      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.{{cite journal}}: CS1 maint: multiple names: authors list (link)

      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. "David Hoffman - the Mathematics Genealogy Project".
      6. Michael, J. H.; Simon, L. M. (1973). "Sobolev and mean-value inequalities on generalized submanifolds of Rn". Communications on Pure and Applied Mathematics. 26 (3): 361–379. doi:10.1002/cpa.3160260305. MR 0344978. Zbl 0256.53006.
      7. Huisken, Gerhard (1986). "Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature". Inventiones Mathematicae. 84 (3): 463–480. Bibcode:1986InMat..84..463H. doi:10.1007/BF01388742. hdl:11858/00-001M-0000-0013-592E-F. MR 0837523. S2CID 55451410. Zbl 0589.53058.
      8. Schoen, Richard; Yau, Shing Tung (1981). "Proof of the positive mass theorem. II". Communications in Mathematical Physics. 79 (2): 231–260. Bibcode:1981CMaPh..79..231S. doi:10.1007/BF01942062. MR 0612249. S2CID 59473203. Zbl 0494.53028.
      9. Meeks, William, III; Simon, Leon; Yau, Shing Tung (1982). "Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature". Annals of Mathematics. Second Series. 116 (3): 621–659. doi:10.2307/2007026. JSTOR 2007026. MR 0678484. Zbl 0521.53007.{{cite journal}}: CS1 maint: multiple names: authors list (link)
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