Joseph Harold Sampson Jr. (1926 – 2003) was an American mathematician known for his work in

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

and

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, especially his work about

harmonic map In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for ...

s in collaboration with

James Eells James Eells (October 25, 1926 – February 14, 2007) was an American mathematician, who specialized in mathematical analysis. Biography Eells studied mathematics at Bowdoin College in Maine and earned his undergraduate degree in 1947. Afte ...

. He obtained his Ph.D. in mathematics from

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 ...

in 1951 under the supervision of

Salomon Bochner Salomon Bochner (20 August 1899 – 2 May 1982) was an Austrian mathematician, known for work in mathematical analysis, probability theory and differential geometry. Life He was born into a Jewish family in Podgórze (near Kraków), then Aus ...

Mathematical work

In 1964, Sampson and

introduced

s, which are mappings between

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

s which solve a geometrically-defined system of

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. They can also be defined via the calculus of variations. Generalizing Bochner's work on

harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \f ...

s, Eells and Sampson derived the

Bochner identity In mathematics — specifically, differential geometry — the Bochner identity is an identity concerning harmonic maps between Riemannian manifolds. The identity is named after the American mathematician Salomon Bochner. Statement of ...

, and used it to prove the triviality of harmonic maps under certain curvature conditions. Eells and Sampson established the existence of harmonic maps whenever the domain manifold is closed and the target has nonpositive

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...

. Their proof analyzed the harmonic map heat flow, which is a geometrically-defined heat equation. By establishing ''a priori estimates'' for the flow, they were able to prove its convergence under the indicated curvature assumption. The use of the Bochner identity in deriving estimates is where the assumption on sectional curvature plays a crucial role. As a result of Eells and Sampson's (subsequential) convergence theorem, they were able to prove the existence of harmonic maps in any

homotopy class In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

. As such, harmonic maps may be regarded as canonically-defined representatives of topological spaces of mappings. This perspective has enabled the application of harmonic maps to many problems in geometry and topology. Eells and Sampson's work is one of the most famous papers in the field of

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 was a direct inspiration for Richard Hamilton's epochal work on the

Ricci flow In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be ana ...

. In addition to Eells and Sampson's heat flow, their main results on existence of harmonic maps can also be derived via the calculus of variations, using the regularity theory developed in the 1980s by

Richard Schoen Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984. Career Born in Celina, Ohio, and a ...

and

Karen Uhlenbeck Karen Keskulla Uhlenbeck (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W. Richard ...

. Later, in 1978, Sampson developed unique continuation,

maximum principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...

s, further rigidity theorems, and deformability results for harmonic maps. He also proved that a harmonic map of degree one between compact hyperbolic

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

s must be a

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...

. The same result was obtained at the same time by Schoen and

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 ...

Major publications

Over the course of forty years, Sampson published around twenty research articles. * *

References

{{DEFAULTSORT:Sampson, Joseph H. American mathematicians 1926 births 2003 deaths Princeton University alumni