Gradient Flows and Łojasiewicz–Simon Estimates in Geometric Analysis

Łojasiewicz-type inequalities give a precise way to turn “almost critical” into quantitative information. Near a critical point of a real-analytic functional, the (gradient) Łojasiewicz inequality bounds energy gap in terms of the size of the gradient. From this follows the quantitative convergence to the unique limit point along the gradient flow. While the classical inequalities apply to finite dimensions, Simon’s work gives the infinite-dimensional extension tailored to applications in geometric analysis. The same perspective also applies when one studies rescalings and limiting models: these inequalities can control the rate at which a flow approaches its tangent flow, or a blowup approaches the tangent cone.
In the lectures we will develop this method in a way that is usable across problems, and then illustrate it in a few representative geometric settings. On the parabolic side, we will treat geometric gradient flows such as harmonic map heat flow and mean curvature flow, focusing on what Łojasiewicz–Simon adds beyond energy decay: quantitative rates, convergence, uniqueness of the limit, regularity of the arrival time, etc. We will also discuss non-parabolic applications including convergence to unique tangent cones of Einstein manifolds and near-equality phenomena in isoperimetric-type inequalities.

Relevance

This course will be most relevant to graduate students and early-stage researchers working in geometry and PDE, especially those interested in long-time behavior of evolution equations and stability/convergence near critical geometric structures.

Pre-requisites

Familiarity with the language of smooth manifolds and Riemannian geometry (metrics, connections, curvature, variation formulas), and standard PDE notions (Sobolev spaces, elliptic regularity) will be assumed. Familiarity with at least one of the specific topics mentioned in the outline (harmonic map heat flow, mean curvature flow, Einstein manifolds, isoperimetric inequality) is helpful but not required.

Pre-Reading

Students are encouraged to review introductory texts in PDE and geometry, such as:

• Evans, Partial Differential Equations (Chapters 2, 5-9); and
• Jost, Riemannian Geometry and Geometric Analysis (Chapters 1, 3-6, optionally 9-10); or Petersen, Riemannian Geometry (chapters 1-5, optionally 6, 7, 11), or similar.