Ngô Quốc Anh

February 23, 2011

Some References for Geometric Evolution Equations

Filed under: Uncategorized — Ngô Quốc Anh @ 16:47

I found this syllabus for a course entitled Geometric Evolution Equations offered in SISSA. According to the material, all topics focus on the flow method in geometric analysis such as

• Mean curvature flow;
• Inverse mean curvature flow;
• Willmore flow;
• Ricci flow.

As can be seen from each topic there are several references that I am going to drop them here for convenience.

1. Parabolic equations
This topic focuses only on the theory of parabolic equations. To my opinion, the following two books are enough to fully understand this type of equations.

– Friedman, Avner. Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964.
– Lieberman, Gary M.. Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996.

2. Mean curvature flow
This topic focuses on the theory of mean curvature flow known as an example of a geometric flow of hypersurfaces in a Riemannian manifold. Intuitively, a family of surfaces evolves under mean curvature flow if the velocity of which a point on the surface moves is given by the mean curvature of the surface.

– Huisken, Gerhard. Flow by mean curvature of convex surfaces into spheres. J. Differential Geom.20 (1984), no. 1, 237-266. [here]
– Huisken, Gerhard; Sinestrari, Carlo. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math.183 (1999), no. 1 , 45-70. [here]
– Zhu, Xi-Ping. Lectures on mean curvature flows. AMS/IP Studies in Advanced Mathematics, 32. American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002.

3. Inverse mean curvature flow
This topic focuses on the theory of inverse mean curvature flow. This is also an example of a geometric flow. Intuitively, a family of surfaces evolves under IMCF if the outward normal speed at which a point on the surface moves is given by the reciprocal of the mean curvature of the surface.

– Gerhardt,Claus. Flow of nonconvex hypersurfaces into spheres. J. Differential  Geom.32 (1990), no. 1, 299-314. [here]
– Huisken, Gerhard; Ilmanen, Tom. The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differential Geom.59 (2001), no. 3 , 353-437. [here]

4. Willmore flow
This is the geometric flow corresponding to the Willmore energy; it is an $L^2$gradient flow. This type of flow can be briefly described as follows. Over a surface $M$ we can define the so-called Willmore energy of $S$

$\displaystyle\mathcal{W} = \int_S H^2 dA-\int_S K dA$

where $H$ is the mean curvature, $K$ is the Gaussian curvature, and $dA$ is the area form of $S$. Then we can define the Willmore flow as the following

$\displaystyle e[{\mathcal{M}}]=\frac{1}{2} \int_{\mathcal{M}} H^2\mathrm{d}A$

where $H$ stands for the mean curvature of the manifold $\mathcal{M}$. Obviously, flow lines satisfy the differential equation

$\displaystyle\partial_t x(t) = -\nabla \mathcal{W}[x(t)]$

where $x$ is a point belonging to the surface.

This flow leads to an evolution problem in differential geometry: the surface $\mathcal{M}$ is evolving in time to follow variations of steepest descent of the energy. Like surface diffusion (mathematics) it is a fourth-order flow, since the variation of the energy contains fourth derivatives.

– Kuwert, Ernst; Schätzle, Reiner. The Willmore flow with small initial energy. J. Differential Geom.57 (2001), no. 3, 409-441. [here]
– Kuwert, Ernst; Schätzle, Reiner. Gradient flow for the Willmore functional. Comm. Anal. Geom.10 (2002), no .  , 307-339. [here]
– Simonett, Gieri. The Willmore flow near spheres. Differential Integral Equations14 (2001), no. 8, 1005-1014. [here]

5. Ricci flow
The Ricci flow is an intrinsic geometric flow (a process which deforms the metric of a Riemannian manifold) in this case in a manner formally analogous to the diffusion of heat, thereby smoothing out irregularities in the metric. The Ricci flow was first introduced by Richard Hamilton in 1981, and is also referred to as the Ricci-Hamilton flow. It plays an important role in Grigori Perelman‘s solution of the Poincaré conjecture, as well as in the proof of the Differentiable sphere theorem by Brendle and Schoen.

Given a Riemannian manifold with metric tensor $g_{ij}$, we can compute the Ricci tensor $R_{ij}$, which collects averages of sectional curvatures into a kind of “trace” of the Riemann curvature tensor. If we consider the metric tensor (and the associated Ricci tensor) to be functions of a variable which is usually called “time” (but which may have nothing to do with any physical time), then the Ricci flow may be defined by the geometric evolution equation

$\displaystyle\partial_t g_{ij}=-2 R_{ij}$.

The normalized Ricci flow makes sense for compact manifolds and is given by the equation

$\displaystyle\partial_t g_{ij}=-2 R_{ij} +\frac{2}{n} R_\mathrm{avg} g_{ij}$

where $R_\mathrm{avg}$ is the average (mean) of the scalar curvature (which is obtained from the Ricci tensor by taking the trace) and $n$ is the dimension of the manifold. This normalized equation preserves the volume of the metric.

The factor of $-2$ is of little significance, since it can be changed to any nonzero real number by rescaling $t$. However the minus sign ensures that the Ricci flow is well defined for sufficiently small positive times; if the sign is changed then the Ricci flow would usually only be defined for small negative times. (This is similar to the way in which the heat equation can be run forwards in time, but not usually backwards in time.) Informally, the Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions.

– Hamilton, Richard S.. Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982),  no. 2, 255-306. [here]
– Hamilton, Richard S.. Four-manifolds with positive curvature operator. J. Differential Geom.24 (1986), no. 2, 153—179. [here]

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