Classical surface theory.Ĭlassical surface theory is the study of isometric immersions of surfaces into Euclidean 3-space.
Our far-reaching generalization of the classical work of Delaunay classified all complete constant mean curvature surfaces admitting a one-parameter group of isometries the new infinite families of such surfaces generated by this work are currently of interest in other areas of surface theory.
#Differential geometry free#
These include the first breakthrough to finiteness in the extension of the classical Bernstein Theorem, the recent proof of the uniqueness of the helicoid as the only non-flat complete embedded simply-connected minimal surface in 3-space, and the first solution of the free boundary problem for polyhedral surfaces, the prototype for Jost’s Theorem.
#Differential geometry series#
Our research on minimal surfaces has produced a series of outstanding results on what have long been recognized as crucial problems for the theory. We have proved this for compact Riemannian spaces with positively pinched curvature and in another direction established that if two compact surfaces of negative curvature and finite area have the same length data for marked closed geodesics then the two surfaces must be isometric. An important problem in the area is the determination of conditions on a compact Riemannian space which ensure the existence of infinitely many geometrically distinct closed geodesics. The global structure of a space may be investigated by the extensive use of geodesics, minimal surfaces and surfaces of constant mean curvature such surfaces are themselves of physical interest (membranes, soap films and soap bubbles). Geodesics, minimal surfaces and constant mean curvature surfaces. Research at Notre Dame covers the following areas at the forefront of current work in geometric analysis and its applications. The striking feature of modern Differential Geometry is its breadth, which touches so much of mathematics and theoretical physics, and the wide array of techniques it uses from areas as diverse as ordinary and partial differential equations, complex and harmonic analysis, operator theory, topology, ergodic theory, Lie groups, non-linear analysis and dynamical systems. Graduate Study in Differential Geometry at Notre Dame
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