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 ...

, the Yamabe flow is an intrinsic

geometric flow In the mathematical field of differential geometry, a geometric flow, also called a geometric evolution equation, is a type of partial differential equation for a geometric object such as a Riemannian metric or an embedding. It is not a term with ...

—a process which deforms the

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...

of a

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 ...

. First introduced by Richard S. Hamilton, Yamabe flow is for noncompact manifolds, and is the negative ''L''²- gradient flow of the (normalized) total

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 ...

, restricted to a given

conformal class In mathematics, conformal geometry is the study of the set of angle-preserving (conformal map, conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space high ...

: it can be interpreted as deforming a Riemannian metric to a conformal metric of constant scalar curvature, when this flow converges. The Yamabe flow was introduced in response to Richard S. Hamilton's own 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 analo ...

and Rick Schoen's solution of the

Yamabe problem The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds: By computing a formula for how the scalar curvatur ...

on manifolds of positive conformal

Yamabe invariant In mathematics, in the field of differential geometry, the Yamabe invariant, also referred to as the sigma constant, is a real number invariant associated to a smooth manifold that is preserved under diffeomorphisms. It was first written down indep ...

Main results

The fixed points of the Yamabe flow are metrics of constant scalar curvature in the given conformal class. The flow was first studied in the 1980s in unpublished notes of Richard Hamilton. Hamilton conjectured that, for every initial metric, the flow converges to a conformal metric of constant scalar curvature. This was verified by Rugang Ye in the locally conformally flat case. Later,

Simon Brendle Simon Brendle (born June 1981) is a German mathematician working in differential geometry and nonlinear partial differential equations. He received his Dr. rer. nat. from Tübingen University under the supervision of Gerhard Huisken (2001). He ...

proved convergence of the flow for all conformal classes and arbitrary initial metrics. The limiting constant-scalar-curvature metic is typically no longer a Yamabe minimizer in this context. While the compact case is settled, the flow on complete, non-compact manifolds is not completely understood, and remains a topic of current research.

Notes

{{reflist Geometric flow