Richard Schoen, 383-T, 725-0854, schoen "at" math
Office hours: Tu 4:00-5:00, F 3:30-4:30 or by appointment
We will begin with some of the classical work on eigenvalues for Riemannian manifolds including the basic upper and lower bounds, the Lichnerowicz theorem, the heat kernel, and connections with isoperimetric quantities such as the Cheeger constant. We will describe a recent method by B. Andrews and J. Clutterbuck using the heat equation to obtain a sharp bound on the fundamental gap and a sharp lower bound for Riemannian metrics (arXiv:1006.1686, arXiv:1204.5079). We will then discuss work from recent papers on the determination of metrics of fixed area on surfaces which maximize the first eigenvalue. A reference for the classical material is chapters III and IV of the book "Lectures on Differential Geometry" by R. Schoen and S. T. Yau. It will be on reserve in the math library.
Classroom and time: Building 420, Room 371, Time: 11-12:15
Week one: Friday (1/11)
Week two: Monday (1/14), Wednesday (1/16), Friday (1/18)
Week three: No class this week
Week four: Monday (1/28), Wednesday (1/30), Friday (2/1)
Week five: Wednesday (2/6), Friday (2/8)
Week six: Wednesday (2/13), Friday (2/15)
Week seven: Wednesday (2/20), Friday (2/22)
Week eight: Wednesday (2/27), Friday (3/1)
Week nine: Monday (3/4)
Week ten: Monday (3/11), Wednesday (3/13), Friday (3/15)
Below are books and papers which are relevant to this course.
"Lectures on Differential Geomtry", R. Schoen, S. T. Yau, Chapters III and IV, This book is on course reserve in the Math Library.
"Eigenvalues in Riemannian Geometry", I. Chavel, This book gives a more complete and systematic treatment of the subject. This book is on course reserve in the Math Library.
Proof of the fundamental gap conjecture, B. Andrews, J. Clutterbuck, Andrews/Clutterbuck 1
Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue, B. Andrews, J. Clutterbuck, Andrews/Clutterbuck 2