David Allen Hoffman
{{short description|American mathematician}}
{{otherpeople|David Hoffman}}
{{distinguish|David A. Huffman}}
{{infobox scientist
|name=David Allen Hoffman
|nationality=American
|education=Stanford University (PhD)
|occupation=Mathematician
|awards=Chauvenet Prize (1990)
}}
David Allen Hoffman is an American mathematician whose research concerns differential geometry. He is an adjunct professor at Stanford University.{{Cite web|url=https://mathematics.stanford.edu/people/david-hoffman|title=David Hoffman | Mathematics|website=mathematics.stanford.edu}} In 1985, together with William Meeks, he proved that Costa's surface was embedded.{{Cite web|url=https://minimal.sitehost.iu.edu/archive/Tori/Tori/Costa/web/index.html|title=Costa Surface|website=minimal.sitehost.iu.edu}} 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."{{Cite web|url=https://www.ams.org/cgi-bin/fellows/fellows.cgi|title=Fellows of the American Mathematical Society|website=American Mathematical Society}} He was awarded the Chauvenet Prize in 1990 for his expository article "The Computer-Aided Discovery of New Embedded Minimal Surfaces".{{Cite web|url=https://www.maa.org/programs-and-communities/member-communities/maa-awards/writing-awards/chauvenet-prizes|title=Chauvenet Prizes | Mathematical Association of America|website=www.maa.org}} He obtained his Ph.D. from Stanford University in 1971 under the supervision of Robert Osserman.{{MathGenealogy|id=24416}}
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.{{cite journal|last1=Michael|first1=J. H.|last2=Simon|mr=0344978|first2=L. M.|zbl=0256.53006|title=Sobolev and mean-value inequalities on generalized submanifolds of {{math|Rn}}|journal=Communications on Pure and Applied Mathematics|volume=26|year=1973|issue=3 |pages=361–379|doi=10.1002/cpa.3160260305|author-link2=Leon Simon}} One year later, Hoffman and Joel Spruck extended Michael and Simon's work to the setting of functions on immersed submanifolds of Riemannian manifolds.{{ran|HS74}} Such inequalities are useful for many problems in geometric analysis which deal with some form of prescribed mean curvature.{{cite journal|last1=Huisken|first1=Gerhard|title=Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature|journal=Inventiones Mathematicae|volume=84|year=1986|issue=3|pages=463–480|mr=0837523|doi=10.1007/BF01388742|bibcode=1986InMat..84..463H |zbl=0589.53058|author-link1=Gerhard Huisken|hdl=11858/00-001M-0000-0013-592E-F|s2cid=55451410 |hdl-access=free}}{{cite journal|last1=Schoen|first1=Richard|author-link1=Richard Schoen|last2=Yau|first2=Shing Tung|title=Proof of the positive mass theorem. II|zbl=0494.53028|journal=Communications in Mathematical Physics|volume=79|year=1981|issue=2|pages=231–260|mr=0612249|url=https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-79/issue-2/Proof-of-the-positive-mass-theorem-II/cmp/1103908964.full|doi=10.1007/BF01942062|bibcode=1981CMaPh..79..231S |s2cid=59473203 |author-link2=Shing-Tung Yau}} As usual for Sobolev inequalities, Hoffman and Spruck were also able to derive new isoperimetric inequalities for submanifolds of Riemannian manifolds.{{ran|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.{{ran|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.{{cite journal|zbl=0521.53007|last1=Meeks|first1=William III|author-link1=William Hamilton Meeks, III|last2=Simon|first2=Leon|author-link2=Leon Simon|last3=Yau|first3=Shing Tung|title=Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature|journal=Annals of Mathematics |series=Second Series|volume=116|year=1982|issue=3|pages=621–659|mr=0678484|doi=10.2307/2007026|jstor=2007026 |author-link3=Shing-Tung Yau}} Hoffman and Meeks' result rules out the latter possibility.
Major publications
{{refbegin}}
- {{rma|HS74|tw=3em|{{cite journal|first1=David|last1=Hoffman|first2=Joel|last2=Spruck|title=Sobolev and isoperimetric inequalities for Riemannian submanifolds|journal=Communications on Pure and Applied Mathematics|volume=27|issue=6|year=1974|pages=715–727|doi=10.1002/cpa.3160270601|author-link2=Joel Spruck|mr=0365424|zbl=0295.53025}} {{erratum|doi=10.1002/cpa.3160280607|checked=yes}}}}
- {{rma|HM90|tw=3em|{{cite journal|first1=D.|last1=Hoffman|first2=W. H. III|last2=Meeks|author-link2=William Hamilton Meeks, III|url=http://www.digizeitschriften.de/dms/resolveppn/?PID{{=}}GDZPPN002107651|title=The strong halfspace theorem for minimal surfaces|journal=Inventiones Mathematicae|volume=101|year=1990|issue=2|pages=373–377|doi=10.1007/bf01231506|bibcode=1990InMat.101..373H |mr=1062966|zbl=0722.53054|s2cid=10695064 |url-access=subscription}}}}
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References
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Category:20th-century American mathematicians
Category:Fellows of the American Mathematical Society
Category:Year of birth missing (living people)
Category:21st-century American mathematicians