David Allen Hoffman

American mathematician

For other people named David Hoffman, see David Hoffman (disambiguation).

Not to be confused with David A. Huffman.

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.[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

[edit]

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

[edit]

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 0295.53025.715-727&rft.date=1974&rft_id=https://zbmath.org/?format=complete&q;=an:0295.53025#id-name=Zbl&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0365424#id-name=MR&rft_id=info:doi/10.1002/cpa.3160270601&rft.aulast=Hoffman&rft.aufirst=David&rft.au=Spruck,+Joel&rfr_id=info:sid/en.wikipedia.org:David+Allen+Hoffman" class="Z3988"> (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 0722.53054.373-377&rft.date=1990&rft_id=https://zbmath.org/?format=complete&q;=an:0722.53054#id-name=Zbl&rft_id=https://api.semanticscholar.org/CorpusID:10695064#id-name=S2CID&rft_id=info:bibcode/1990InMat.101..373H&rft_id=info:doi/10.1007/bf01231506&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=1062966#id-name=MR&rft.aulast=Hoffman&rft.aufirst=D.&rft.au=Meeks,+W.+H.+III&rft_id=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002107651&rfr_id=info:sid/en.wikipedia.org:David+Allen+Hoffman" class="Z3988">

|

## References

[edit]

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.Rn&rft.volume=26&rft.issue=3&rft.pages=361-379&rft.date=1973&rft_id=https://zbmath.org/?format=complete&q;=an:0256.53006#id-name=Zbl&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0344978#id-name=MR&rft_id=info:doi/10.1002/cpa.3160260305&rft.aulast=Michael&rft.aufirst=J.+H.&rft.au=Simon,+L.+M.&rfr_id=info:sid/en.wikipedia.org:David+Allen+Hoffman" class="Z3988">
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 0589.53058.463-480&rft.date=1986&rft_id=https://zbmath.org/?format=complete&q;=an:0589.53058#id-name=Zbl&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0837523#id-name=MR&rft_id=info:bibcode/1986InMat..84..463H&rft_id=info:doi/10.1007/BF01388742&rft_id=https://api.semanticscholar.org/CorpusID:55451410#id-name=S2CID&rft_id=info:hdl/11858/00-001M-0000-0013-592E-F&rft.aulast=Huisken&rft.aufirst=Gerhard&rfr_id=info:sid/en.wikipedia.org:David+Allen+Hoffman" class="Z3988">
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 0494.53028.231-260&rft.date=1981&rft_id=https://zbmath.org/?format=complete&q;=an:0494.53028#id-name=Zbl&rft_id=https://api.semanticscholar.org/CorpusID:59473203#id-name=S2CID&rft_id=info:bibcode/1981CMaPh..79..231S&rft_id=info:doi/10.1007/BF01942062&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0612249#id-name=MR&rft.aulast=Schoen&rft.aufirst=Richard&rft.au=Yau,+Shing+Tung&rft_id=https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-79/issue-2/Proof-of-the-positive-mass-theorem-II/cmp/1103908964.full&rfr_id=info:sid/en.wikipedia.org:David+Allen+Hoffman" class="Z3988">
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 0521.53007.621-659&rft.date=1982&rft_id=https://zbmath.org/?format=complete&q;=an:0521.53007#id-name=Zbl&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0678484#id-name=MR&rft_id=https://www.jstor.org/stable/2007026#id-name=JSTOR&rft_id=info:doi/10.2307/2007026&rft.aulast=Meeks&rft.aufirst=William+III&rft.au=Simon,+Leon&rft.au=Yau,+Shing+Tung&rfr_id=info:sid/en.wikipedia.org:David+Allen+Hoffman" class="Z3988">

Authority control databases

International|

* ISNI
* VIAF
* WorldCat

National|

* Germany
* United States
* France
* BnF data
* Netherlands
* Israel

Academics|

* Mathematics Genealogy Project
* zbMATH
* MathSciNet

Other|

* IdRef