Shiu-Yuen Cheng (鄭紹遠) is a

Hong Kong Hong Kong ( (US) or (UK); , ), officially the Hong Kong Special Administrative Region of the People's Republic of China ( abbr. Hong Kong SAR or HKSAR), is a city and special administrative region of China on the eastern Pearl River Delt ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

. He is currently the Chair Professor of Mathematics at the

Hong Kong University of Science and Technology The Hong Kong University of Science and Technology (HKUST) is a public research university in Clear Water Bay Peninsula, New Territories, Hong Kong. Founded in 1991 by the British Hong Kong Government, it was the territory's third institution ...

. Cheng received his Ph.D. in 1974, under the supervision of

Shiing-Shen Chern Shiing-Shen Chern (; , ; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geome ...

, from

University of California at Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant univ ...

. Cheng then spent some years as a post-doctoral fellow and assistant professor at

Princeton University Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...

and the

State University of New York at Stony Brook Stony Brook University (SBU), officially the State University of New York at Stony Brook, is a public research university in Stony Brook, New York. Along with the University at Buffalo, it is one of the State University of New York system's ...

. Then he became a full professor at

University of California at Los Angeles The University of California, Los Angeles (UCLA) is a public university, public Land-grant university, land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a Normal school, teachers colle ...

. Cheng chaired the Mathematics departments of both the

Chinese University of Hong Kong The Chinese University of Hong Kong (CUHK) is a public research university in Ma Liu Shui, Hong Kong, formally established in 1963 by a charter granted by the Legislative Council of Hong Kong. It is the territory's second-oldest university an ...

and the

in the 1990s. In 2004, he became the Dean of Science at HKUST. In 2012, he became a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

. He is well known for contributions to

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

and

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

s, including Cheng's eigenvalue comparison theorem, Cheng's maximal diameter theorem, and a number of works with

Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...

. Many of Cheng and Yau's works formed part of the corpus of work for which Yau was awarded the

Fields medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...

in 1982. As of 2020, Cheng's most recent research work was published in 1996.

Technical contributions

Gradient estimates and their applications

In 1975,

found a novel gradient estimate for solutions of second-order

elliptic partial differential equation Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, wher ...

s on certain complete Riemannian manifolds. Cheng and Yau were able to localize Yau's estimate by making use of a method developed by

Eugenio Calabi Eugenio Calabi (born 11 May 1923) is an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics, Emeritus, at the University of Pennsylvania, specializing in differential geometry, partial differential equations and ...

. The result, known as the Cheng–Yau gradient estimate, is ubiquitous in the field of

geometric analysis Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of l ...

. As a consequence, Cheng and Yau were able to show the existence of an eigenfunction, corresponding to the first eigenvalue, of the Laplace-Beltrami operator on a complete Riemannian manifold. Cheng and Yau applied the same methodology to understand spacelike hypersurfaces of

Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...

and the geometry of hypersurfaces in

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...

. A particular application of their results is a Bernstein theorem for closed spacelike hypersurfaces of Minkowski space whose mean curvature is zero; any such hypersurface must be a plane. In 1916,

Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...

found a differential identity for the geometric data of a convex surface in Euclidean space. By applying the maximum principle, he was able to control the extrinsic geometry in terms of the intrinsic geometry. Cheng and Yau generalized this to the context of hypersurfaces in Riemannian manifolds.

The Minkowski problem and the Monge-Ampère equation

Any strictly convex closed hypersurface in the

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

can be naturally considered as an embedding of the -dimensional sphere, via the

Gauss map In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that ' ...

. The

Minkowski problem In differential geometry, the Minkowski problem, named after Hermann Minkowski, asks for the construction of a strictly convex compact surface ''S'' whose Gaussian curvature is specified. More precisely, the input to the problem is a strictly posit ...

asks whether an arbitrary smooth and positive function on the -dimensional sphere can be realized as the

scalar curvature In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...

of the

Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...

induced by such an embedding. This was resolved in 1953 by

Louis Nirenberg Louis Nirenberg (February 28, 1925 – January 26, 2020) was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century. Nearly all of his work was in the field of partial differential equat ...

, in the case that is equal to two. In 1976, Cheng and Yau resolved the problem in general. By the use of the

Legendre transformation In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions of ...

, solutions of the Monge-Ampère equation also provide convex hypersurfaces of Euclidean space; the scalar curvature of the intrinsic metric is prescribed by the right-hand sided of the Monge-Ampère equation. As such, Cheng and Yau were able to use their resolution of the Minkowski problem to obtain information about solutions of Monge-Ampère equations. As a particular application, they obtained the first general existence and uniqueness theory for the boundary-value problem for the Monge-Ampère equation.

Luis Caffarelli Luis Angel Caffarelli (born December 8, 1948) is an Argentine mathematician and luminary in the field of partial differential equations and their applications. Career Caffarelli was born and grew up in Buenos Aires. He obtained his Masters of S ...

, Nirenberg, and

Joel Spruck Joel Spruck (born 1946) is a mathematician, J. J. Sylvester Professor of Mathematics at Johns Hopkins University, whose research concerns geometric analysis and elliptic partial differential equations. He obtained his PhD from Stanford University ...

later developed more flexible methods to deal with the same problem.L. Caffarelli, L. Nirenberg, and J. Spruck. The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Comm. Pure Appl. Math. 37 (1984), no. 3, 369–402.

Major publications

References

External links

School of Science, the Hong Kong University of Science and TechnologyMathematics Department, the Hong Kong University of Science and Technology
Living people Hong Kong mathematicians Hong Kong University of Science and Technology faculty Fellows of the American Mathematical Society Year of birth missing (living people) {{asia-mathematician-stub