Shiu-Yuen Cheng (鄭紹遠) is a

Hong Kong Hong Kong)., Legally Hong Kong, China in international treaties and organizations. is a special administrative region of China. With 7.5 million residents in a territory, Hong Kong is the fourth most densely populated region in the wor ...

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, mathematical structure, structure, space, Mathematica ...

. 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 Sai Kung District, New Territories, Hong Kong. Founded in 1991, it was the territory's third institution to be granted university status, and the firs ...

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

Shiing-Shen Chern Shiing-Shen Chern (; , ; October 26, 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, United States. Founded in 1868 and named after the Anglo-Irish philosopher George Berkele ...

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

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

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 university, public research university in Stony Brook, New York, United States, on Long Island. Along with the University at Buffalo, it is on ...

. Then he became a full professor at

University of California at Los Angeles The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California, United States. Its academic roots were established in 1881 as a normal school then known as the southern branch of the Ca ...

. Cheng chaired the Mathematics departments of both the

Chinese University of Hong Kong The Chinese University of Hong Kong (CUHK) is a public university, public research university in Sha Tin, New Territories, Hong Kong. Established in 1963 as a federation of three university college, collegesChung Chi College, New Asia Coll ...

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 Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

and

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

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. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...

. 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 Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...

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 In mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDEs are frequently used to model steady states, unlike parabolic PDE and hyperbolic PDE which gene ...

s on certain complete

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

s. Cheng and Yau were able to localize Yau's estimate by making use of a method developed by Eugenio Calabi. 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 ...

. 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 physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a ...

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 relat ...

. 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, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...

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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

can be naturally considered as an embedding of the -dimensional sphere, via the Gauss map. The Minkowski problem 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 is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

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 Mathematical analysis, mathematicians of the 20th century. Nearly all of his work was in the field of par ...

, 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), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued functions that are convex on a rea ...

, 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, Nirenberg, and Joel Spruck 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 Academic staff of the Hong Kong University of Science and Technology Fellows of the American Mathematical Society Year of birth missing (living people) {{asia-mathematician-stub