# Shing-Tung Yau

**Shing-Tung Yau** (/jaʊ/; Chinese: 丘成桐; pinyin: *Qiū Chéngtóng*; Cantonese Yale: *Yāu Sìngtùng*; born April 4, 1949) is a Chinese-born naturalized American mathematician who was awarded the Fields Medal in 1982. He is currently the William Caspar Graustein Professor of Mathematics at Harvard University.[1]

Shing-Tung Yau | |
---|---|

Born | |

Nationality | United States |

Alma mater | Chinese University of Hong Kong (B.A. 1969) University of California, Berkeley (Ph. D 1971) |

Known for | Calabi conjecture Calabi–Yau manifold Positive energy theorem SYZ conjecture Yau's conjecture |

Spouse(s) | Yu-yun Kuo |

Children | 2 Sons |

Awards | John J. Carty Award (1981) Veblen Prize (1981) Fields Medal (1982) Crafoord Prize (1994) National Medal of Science (1997) Wolf Prize (2010) |

Scientific career | |

Fields | Mathematics |

Institutions | Harvard University Stanford University Stony Brook University Institute for Advanced Study |

Doctoral advisor | Shiing-Shen Chern |

Doctoral students | Richard Schoen (Stanford, 1977) Jun Li (Stanford, 1989) Huai-Dong Cao (Princeton, 1986) Gang Tian (Harvard, 1988) Lizhen Ji (Northeastern, 1991) Kefeng Liu (Harvard, 1993) Mu-Tao Wang (Harvard, 1998) Chiu-Chu Melissa Liu (Harvard, 2002) Mark Stern (Princeton, 1984) |

Yau's work is mainly in differential geometry, especially in geometric analysis. His contributions have influenced both physics and mathematics, and he has been active at the interface between geometry and theoretical physics. His proof of the positive energy theorem in general relativity demonstrated—sixty years after its discovery—that Einstein's theory is consistent and stable. His proof of the Calabi conjecture allowed physicists to show, using Calabi–Yau compactification, that string theory is a viable candidate for a unified theory of nature. Calabi–Yau manifolds are part of the standard toolkit for string theorists today.

## Biography

Yau was born in Shantou, Guangdong, China with Hakka ancestry in Jiaoling County. He has seven siblings, including Stephen Shing-Toung Yau, also a mathematician.[2] When he was only a few months old, his family moved to Hong Kong.

Yau's father, Yau Chenying, was a patriotic Chinese philosophy professor who worked against the invading Japanese. Under the influence of his father, Yau acquired broad knowledge of classical Chinese literature and history, which resulted in an essay *On Mathematics and Chinese literature* (數學和中國文學的比較), with reference to *Dream of the Red Chamber* and Wang Guowei, explaining the structural relationship between mathematics and Chinese literature, published in 2006. His mother came from Mei County.

After graduating from Pui Ching Middle School, he studied mathematics at the Chinese University of Hong Kong from 1966 to 1969. Yau left for the University of California, Berkeley in the fall of 1969, where he received his Ph.D. in mathematics two years later, under the supervision of Shiing-Shen Chern. He spent a year as a member of the Institute for Advanced Study at Princeton before joining Stony Brook University in 1972 as an assistant professor. In 1974, he became an associate professor at Stanford University.[3] In 1978, Yau became stateless after the British Consulate revoked his Hong Kong residency due to his United States permanent residency status.[4] Yau remained stateless until 1990, when he obtained United States citizenship.[4][5]

From 1984 to 1987 he worked at University of California, San Diego.[6] Since 1987, he has been at Harvard University.[7]

## Contributions to mathematics

### Work on conjectures

#### Calabi conjecture

Yau's solution of the Calabi conjecture, concerning the existence of an Einstein–Kähler metric, has far-reaching consequences. The existence of such a canonical unique metric allows one to give explicit representatives of characteristic classes. Calabi–Yau manifolds are now fundamental in string theory, where the Calabi conjecture provides an essential piece in the model.

In algebraic geometry, the Calabi conjecture implies the Miyaoka–Yau inequality on Chern numbers of surfaces, a characterization of the complex projective plane and quotients of the two-dimensional complex unit ball, an important class of Shimura varieties.

Yau also made a contribution in the case that the first Chern number *c*_{1} > 0, and conjectured its relation to the stability in the sense of geometric invariant theory in algebraic geometry. This has motivated the work of Simon Donaldson on scalar curvature and stability. Another important result of Donaldson–Uhlenbeck–Yau is that a holomorphic vector bundle is stable (in the sense of David Mumford) if and only if there exists an Hermitian–Yang–Mills metric on it. This has many consequences in algebraic geometry, such as the characterization of certain symmetric spaces, Chern number inequalities for stable bundles, and the restriction of the fundamental groups of a Kähler manifold.

#### Positive mass conjecture and existence of black holes

Yau pioneered the method of using minimal surfaces to study geometry and topology. By analyzing how minimal surfaces behave in space-time, Yau and Richard Schoen proved the long-standing conjecture that the total mass in general relativity is positive.

This theorem implies that flat space-time is stable, a fundamental issue for the theory of general relativity. Briefly, the positive mass conjecture says that if a three-dimensional manifold has positive scalar curvature and is asymptotically flat, then a constant that appears in the asymptotic expansion of the metric is positive. A continuation of the above work leads to another result in relativity proved by Yau, an existence theorem for black holes that extends and strengthens earlier results of Penrose and Hawking. Yau and Schoen continued their work on manifolds with positive scalar curvature, which led to Schoen's final solution of the Yamabe problem.

#### Smith conjecture

Yau and William H. Meeks resolved the well-known question of whether the Douglas solution of a minimal disk for an external Jordan curve, the Plateau problem, in three space, is always embedded if the boundary curve is a subset of a convex boundary. They then went on to prove that these embedded minimal surfaces are equivariant for finite group actions. Combining this work with a result by William Thurston, Cameron Gordon assembled a proof of the Smith conjecture: for any cyclic group acting on a sphere, the set of fixed points is not a knotted curve.

#### Hermitian Yang–Mills connection and stable vector bundles

Yau and Karen Uhlenbeck proved the existence and uniqueness of Hermitian–Einstein metrics (or equivalently Hermitian Yang–Mills connections) for stable bundles on any compact Kähler manifold, extending an earlier result of Donaldson for projective algebraic surfaces, and M. S. Narasimhan and C. S. Seshadri for algebraic curves. Both the results and methods of this paper have been influential in parts of both algebraic geometry and string theory. This result is now usually called the Donaldson–Uhlenbeck–Yau theorem.

#### Frankel conjecture

Yau and Yum-Tong Siu proved the 1981 Frankel conjecture in complex geometry, stating that any compact positively curved Kähler manifold is biholomorphic to complex projective space. An independent proof was given by Shigefumi Mori, using methods of algebraic geometry in positive characteristic.

#### Mirror conjecture

With Bong Lian and Kefeng Liu, Yau proved the mirror formulas conjectured by string theorists in a paper posted in December 1997.[8][9] These formulas give the explicit numbers of rational curves of all degrees in a large class of Calabi–Yau manifolds, in terms of the Picard–Fuchs equations of the corresponding mirror manifolds. Alexander Givental had earlier sketched a proof of the mirror formulas in a paper on "Equivariant Gromov-Witten invariants" in March 1996 [10] (Internat. Math. Res. Notices 1996).

### New methods and concepts

#### Gradient estimates and Harnack inequalities

Yau developed the method of gradient estimates for Harnack's inequalities. This method has been used and refined by Yau and other mathematicians to attack, for example, bounds on the heat kernel. Early in 1981, Yau suggested to Richard S. Hamilton that he use the Ricci flow to realize naturally the canonical decomposition of a three-dimensional manifold into pieces, each of which has a geometric structure, in the Thurston program. Hamilton amplified their results, leading to the Li–Yau–Hamilton inequality for the Ricci flow equations.

Gradient estimates were also used in Yau's joint work with Shiu-Yuen Cheng to give a complete proof of the higher-dimensional Hermann Minkowski problem and the Dirichlet problem for the real Monge–Ampère equation, and other results on the Kähler–Einstein metric of bounded pseudoconvex domains.

#### Uniformization of complex manifolds

When Yau was a graduate student, he started to generalize the uniformization theorem of Riemann surfaces to higher-dimensional complex Kähler manifolds. For a compact manifold with positive bisectional curvature, the Frankel conjecture - proved by Siu and Yau, and independently by Mori - shows that it is complex projective space. Yau proposed a series of conjectures when the manifold is non-compact, and made contributions towards their solutions. For example, when the bisectional curvature is positive, it must be biholomorphic to *C*^{n}.

#### Harmonic maps and rigidity

When Yau was working on his thesis about manifolds with non-positive curvature and their fundamental groups, he realized that it is possible to use harmonic maps to give alternative proofs of some results there. He was aware of the Mostow rigidity theorem for locally symmetric spaces, which he used to prove the uniqueness of complex structure of quotients of complex balls. He proposed that harmonic maps be used to prove rigidity of the complex structure for Kähler manifolds with strongly negative curvature, a program that was successfully carried out by Yum-Tong Siu. This method, the Siu-Yau method, has been extended to prove strong and super-rigidities of many locally symmetric spaces.

#### Minimal submanifolds

Yau has used minimal submanifolds in his solutions of the Positive Mass Conjecture, the Smith conjecture, and the Frankel conjecture, among others. Many other mathematicians have since applied minimal surfaces to other problems. Mikhail Gromov's introduction of pseudo-holomorphic curves in symplectic geometry has also had an important impact on this field.

### Open problems

Yau has compiled an influential set of open problems in geometry.

#### Existence of minimal surfaces

It is known that any closed surface possesses infinitely many closed geodesics. The first problem in the minimal submanifolds section of Yau's list asks whether any closed three-manifold has infinitely many closed smooth immersed minimal surfaces. At the time it was known from Almgren–Pitts min-max theory the existence of at least one minimal surface. Kei Irie, Fernando Codá Marques, and André Neves solved this problem in the generic case [11] and later Antoine Song claimed it in full generality.[12]

#### Harmonic functions with controlled growth

One of Yau's problems is about bounded harmonic functions, and harmonic functions on noncompact manifolds of polynomial growth. After proving non-existence of bounded harmonic functions on manifolds with positive curvatures, he proposed the Dirichlet problem at infinity for bounded harmonic functions on negatively curved manifolds, and then proceeded to harmonic functions of polynomial growth. Dennis Sullivan tells a story about Yau's geometric intuition, and how it led him to reject an analytical proof of Sullivan's. Michael Anderson independently found the same result about bounded harmonic function on simply connected negatively curved manifolds using a geometric convexity construction.[13]

#### Rank rigidity of nonpositively curved manifolds

Again motivated by Mostow's strong rigidity theorem, Yau called for a notion of rank for general manifolds extending the one for locally symmetric spaces, and asked for rigidity properties for higher rank metrics. Advances in this direction have been made by Ballmann, Brin and Eberlein in their work on non-positive curved manifolds, Gromov's and Eberlein's metric rigidity theorems for higher rank locally symmetric spaces and the classification of closed higher rank manifolds of non-positive curvature by Ballmann and Burns-Spatzier. This leaves rank 1 manifolds of non-positive curvature as the focus of research. They behave more like manifolds of negative curvature, but remain poorly understood in many regards.

#### Kähler–Einstein metrics and stability of manifolds

It is known that if a complex manifold has a Kähler–Einstein metric, then its tangent bundle is stable. Yau realized early in 1980s that the existence of special metrics on Kähler manifolds is equivalent to the stability of the manifolds. Various people including Simon Donaldson have made progress to understand such a relation.

#### Mirror symmetry

He has collaborated with string theorists including Strominger, Vafa and Witten, and as post-doctorals from theoretical physics with Brian Greene, Eric Zaslow and A. Klemm . The Strominger–Yau–Zaslow program is to construct explicitly mirror manifolds. David Gieseker wrote of the seminal role of the Calabi conjecture in relating string theory with algebraic geometry, in particular for the developments of the SYZ program, mirror conjecture and Yau–Zaslow conjecture.[13]

## Initiatives in mainland China and Taiwan

After China entered the reform and opening era, Yau revisited China in 1979 on the invitation of Hua Luogeng.

To help develop Chinese mathematics, Yau started by educating students from China. He then began establishing mathematics research institutes and centers, organizing conferences at all levels, initiating out-reach programs, and raising private funds for these purposes. John Coates has commented on Yau's success as a fundraiser.[13] The first of Yau's initiatives is The Institute of Mathematical Sciences at The Chinese University of Hong Kong in 1993. The goal is to "organize activities related to a broad variety of fields including both pure and Applied mathematics, scientific computation, image processing, mathematical physics and statistics. The emphasis is on interaction and linkages with the physical sciences, engineering, industry and commerce."

Yau's second major initiative is the Morningside Center of Mathematics in Beijing, established in 1996. Part of the money for the building and regular operations was raised by Yau from the Morningside Foundation in Hong Kong. Yau also proposed organizing the International Congress of Chinese Mathematicians, which is now held every three years. The first congress was held at the Morningside Center from December 12 to 18, 1998.

His third initiative is the Center of Mathematical Sciences at Zhejiang University, established in 2002. Yau is the director of all three mathematics institutes and visits them on a regular basis.

Yau went to Taiwan to attend a conference in 1985. In 1990, he was invited by Liu Chao-shiuan, then the President of National Tsinghua University, to visit the university for a year. A few years later, he convinced Liu, then-chairman of National Science Council, to create the National Center of Theoretical Sciences (NCTS), which was established at Hsinchu in 1998. He was the chairman of the Advisory Board of the NCTS until 2005.

## Outreach

His classmate at college Y.-C.Siu speaks of Yau as an ambassador of mathematics.[13] In Hong Kong, with the support of Ronnie Chan, Yau set up the Hang Lung Award for high school students. He has also organized and participated in meetings for high school and college students, such as the panel discussions *Why Math? Ask Masters!* in Hangzhou, July 2004, and *The Wonder of Mathematics* in Hong Kong, December 2004. Yau organized the JDG conference surveying developments in geometry and related fields, in addition to the annual Current development of mathematics conference. Yau also co-initiated a series of books on popular mathematics, "Mathematics and Mathematical People".

## Honors and awards

Yau has received numerous honors and awards,[14] including:

### Prizes and awards

- 1979, California Scientist of the Year.
- 1981, Oswald Veblen Prize in Geometry.
- 1981, John J. Carty Award for the Advancement of Science, United States National Academy of Sciences.[15]
- 1982, Fields Medal, for "his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge–Ampère equations."
- 1984,
*Science Digest*, America's 100 Brightest Scientists under 40. - 1991, Humboldt Research Award, Alexander von Humboldt Foundation, Germany.
- 1994, Crafoord Prize.[16]
- 1997, United States National Medal of Science.
- 2003, China International Scientific and Technological Cooperation Award, for "his outstanding contribution to PRC in aspects of making progress in sciences and technology, training researchers."
- 2010, Wolf Prize in Mathematics, for "his work in geometric analysis and mathematical physics".[17]
- 2018, Marcel Grossmann Awards, for "for the proof of the positivity of total mass in the theory of general relativity and perfecting as well the concept of quasi-local mass, for his proof of the Calabi conjecture, for his continuous inspiring role in the study of black holes physics."[18]

### Research fellowships

- 1975–1976, Sloan Fellow.
- 1982, Guggenheim Fellowship.
- 1984–1985, MacArthur Fellow.

### Honorary professorships

- Honorary Professor, Hunan Normal University (appointed on Dec 22, 2009).
- Honorary Professor, Northwest University (appointed on Jul 15, 2009).[19]
- Honorary Professor, North University of China (appointed on Jun 18, 2009).[20]
- Honorary Professor, Huazhong University of Science and Technology (appointed on Jan 15, 2006).[21]
- Honorary Professor, University of Science and Technology of China (appointed in 1999).
- Honorary Professor, Peking University (appointed in 1998).
- Honorary Professor, Zhejiang University (appointed in 1998).
- Honorary Professor, Nankai University (appointed in 1993).
- Honorary Professor, Tsinghua University (appointed in 1987).
- Honorary Professor, Hangzhou University (appointed in 1987).
- Honorary Professor, Fudan University (appointed in 1983).

### Honorary degrees

- Honorary Degree of Doctor of Science, The Chinese University of Hong Kong (1980)
- Honorary Master of Arts, Harvard University (1987)
- Honorary Degree of Doctor of Science, National Chiao Tung University, Taiwan (1997)
- Honorary Degree of Doctor of Science, National Tsing Hua University, Taiwan (2000)
- Doctor of Science
*honoris causa*, The University of Macau (2002)[22] - Honorary Doctorate, Zhejiang University (Mar 25, 2003).[23]
- Honorary Degree of Doctor of Science, National Central University (Jul 9, 2004)[24]
- Doctor of Science
*honoris causa*, The Hong Kong University of Science and Technology (Nov, 2004)[25] - Doctor of Science, Polytechnic University in Brooklyn, New York (2005)
- Doctor of Science, National Taiwan University (2005)
- Doctor of Science, Lehigh University (2009)
- Doctor of Science, National Cheng Kung University (2010)
- Doctor of Mathematics, University of Waterloo, Canada (Jun 17, 2011)
- Doctor of Humane Letters, University of Illinois at Chicago (May 11, 2014)

### Academic memberships

- Foreign Member, Indian National Academy of Science (elected in 2008).
- Foreign Member, National Academy of Lincei of Italy (elected in 2005).
- Foreign Member, Russian Academy of Sciences (elected in 2003).
- Foreign member, Chinese Academy of Sciences (elected in 1994).
- Member, United States National Academy of Science (elected in 1993).
- Academician, Academic Sinica (elected in 1984).
- Member, American Academy of Arts and Sciences (elected in 1982).
- Honorary Member, Academic Committee of the Institute of Mathematics, Chinese Academy of Sciences (elected in 1980).
- Fellow, American Association for the Advancement of Science.
- Fellow, Society for Industrial and Applied Mathematics.
- Fellow, American Physical Society.
- Member, Boston Academy of Arts and Sciences.
- Member, New York Academy of Science.
- Honorary Fellow, Shaw College of The Chinese University of Hong Kong.[26]
- Fellow, American Mathematical Society (elected in 2012)[27]

## Poincaré conjecture controversy

In August 2006, a *New Yorker* article, *Manifold Destiny*, alleged that Yau was downplaying Grigori Perelman's work on the Poincaré conjecture.[28] Yau claimed that this article was defamatory, and threatened a lawsuit. *The New Yorker* stood by the story and no lawsuit was filed. In September 2006, Yau established a public relations website, which disputed points in it. Seventeen mathematicians, including two quoted in the *New Yorker* article, posted letters of strong support.[29]

On October 17, 2006, a more sympathetic profile of Yau appeared in *The New York Times*.[30] It devoted about half its length to the Perelman affair. The article stated that Yau had alienated some colleagues, but represented Yau's position as that Perelman's proof was not generally understood and he "had a duty to dig out the truth of the proof".[31]

## Publications

- Yau, Shing Tung (1978), "On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I",
*Communications on Pure and Applied Mathematics*,**31**(3): 339–411, Bibcode:1979CPApM..32..313E, doi:10.1002/cpa.3160310304, MR 0480350 - 2010. (with Steve Nadis)
*The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions*. Basic Books. ISBN 978-0-465-02023-2. - 2019. (with Steve Nadis)
*The Shape of a Life: One Mathematician's Search for the Universe's Hidden Geometry*. Yale University Press. ISBN 978-0-300-23590-6.

## See also

## References

- "Questions and answers with Shing-Tung Yau",
*Physics Today*, 11 April 2016. - "丘成桐院士关注家乡蕉岭仓海诗廊文化建设项目".
*Eastday*(in Chinese). 2018-06-06. Retrieved 2019-08-17. - "Shing-Tung Yau (Biography)".
- Yau, Shing-Tung; Nadis, Steve (2019).
*The Shape of a Life: One Mathematician's Search for the Universe's Hidden Geometry*. Yale University Press. p. 125.Stephen Hawking invited me to discuss [the proof] with him at Cambridge University in late August 1978. I gladly accepted.... Travel was difficult, however, because the British Consulate had recently taken my Hong Kong resident card, maintaining that I could not keep it now that I had a U.S. green card. In the process, I had become stateless. I was no longer a citizen of any country.... until I became a U.S. citizen in 1990.

- Overbye, Dennis (October 17, 2006). "Scientist at Work: Shing-Tung Yau The Emperor of Math".
*The New York Times*. Retrieved September 14, 2013.He became a United States citizen in 1990.

- "University of California, San Diego: External Relations: News & Information: News Releases : Science".
- "Department of Mathematics faculty, Harvard University".
- Lian, B.; Liu, K.; Yau, S. T. (1997-12-11). "Mirror Principle I". arXiv:alg-geom/9712011.
- "On the Mirror Conjecture" (PDF).
- Givental, Alexander B. (1996-03-26). "Equivariant Gromov - Witten Invariants". arXiv:alg-geom/9603021.
- "Density of minimal hypersurfaces for generic metrics | Annals of Mathematics".
- Song, Antoine (2018). "Existence of infinitely many minimal hypersurfaces in closed manifolds". arXiv:1806.08816 [math.DG].
- Page at
*Center of Mathematical Sciences at Zhejiang University* - "YAU, SHING-TUNG" (php). International Center for Scientific Research. Retrieved Jan 1, 2009.
- "John J. Carty Award for the Advancement of Science". United States National Academy of Sciences. Archived from the original on 2010-12-29. Retrieved Jan 1, 2009.
- "...for his development of non-linear techniques in differential geometry leading to the solution of several outstanding problems."
- Malkah Fleisher, Winners of Prestigious Wolf Prize Announced
- Marcel Grossmann, 15th Marcel Grossmann Meeting
- 丘成桐受聘西北大学名誉教授 (shtml) (in Chinese). Chinese Academy of Sciences. 2009-07-15. Retrieved Jan 1, 2009.
- 著名数学家丘成桐受聘我校荣誉教授 (in Chinese). North University of China. 2009-06-18. Archived from the original (shtml) on 2011-07-07. Retrieved Jan 1, 2009.
- 靖咏安 (2006-01-15). 中科院外籍院士丘成桐受聘我校名誉教授 (in Chinese). Huazhong University of Science and Technology. Archived from the original (shtml) on 2011-07-23. Retrieved Jan 1, 2009.
- "Honorary Degrees and Titles" (PDF) (in Chinese and English). The University of Macau. August 9, 2009. Retrieved January 1, 2009.
- 著名数学家丘成桐获浙江大学名誉博士学位 (in Chinese). Zhejiang University News Agency. 2003-03-25. Archived from the original (php) on 2011-07-21. Retrieved Jan 1, 2009.
- 國立中央大學名譽博士頒授 (in Chinese). National Central University. Archived from the original on 2009-07-01. Retrieved Jan 1, 2004.
- "Distinguished Leaders to be Awarded Honorary Doctorates" (PDF). The Hong Kong University of Science and Technology (HKUST). August 26, 2004. Retrieved Jan 1, 2009.
- "Colleges" (PDF) (in Chinese and English). Information Services Office, The Chinese University of Hong Kong. October 15, 2007. Retrieved January 1, 2009.
- List of Fellows of the American Mathematical Society, retrieved 2013-09-01.
- Manifold Destiny: A legendary problem and the battle over who solved it. Retrieved 8 March 2015.
- Yau's website, with information on his legal action and letter to The New Yorker
- Dennis Overbye (17 October 2006). "Shing-tung Yau: The Emperor of Math".
*New York Times*. - Famous scientist slams academic corruption in China Archived 2008-09-17 at the Wayback Machine, China View (Xinhua), 17 August 2006. Retrieved on 2008-08-05.

## External links

*Center of Mathematical Sciences at Zhejiang University*: A large collection of quotes extolling the influence of Yau- Discover Magazine Interview, June 2010 issue
- Interview (11 pages long in Traditional Chinese)
- Yau's autobiographical account (mostly English, some Chinese)
- O'Connor, John J.; Robertson, Edmund F., "Shing-Tung Yau",
*MacTutor History of Mathematics archive*, University of St Andrews. - Shing-Tung Yau at the Mathematics Genealogy Project
- Banquet photos
- Yau's website, with information on his legal action and letter to The New Yorker
- Yau's Public Relations agent, Spector and Associates
- Richard S Hamilton' Letter to Yau Shing-Tung' Attorney on September 25, 2006
- Plugging A Math Gap
- UC Irvine courting Yau with a $2.5 million professorship
- International Conference Celebrating Shing Tung Yau's Birthday 8/27/2008-9/1/2008 Harvard University