## Notes of Andrea Malchiodi’s lecture

A positive mass theorem for CR manifolds

Joint with J H Cheng and P. Yang (Princeton).

1. Asymptotically flat Riemannian manifolds

This is ${{\mathbb R}^3}$ equipped with a metric of the form

$\displaystyle \begin{array}{rcl} g=(1+\frac{M}{r})\delta + h \end{array}$

where first and second derivatives of ${h}$ decay like ${r^{-2-\ell}}$, ${\ell=1,2}$.

1.1. Einstein’s equations

$\displaystyle \begin{array}{rcl} E:=Ric-\frac{1}{2}Rg=T, \end{array}$

where ${T}$ is determined by the distribution of matter. In the vacuum (${T=0}$), Einstein’s equations are variational : stationary points of the total scalar curvature functional.

1.2. Mass

Under non compacty supported variations of the metric, the variation of total scalar curvature involves a boundary term

$\displaystyle \begin{array}{rcl} m(g):=\lim_{r\rightarrow\infty}\int_{S_r}(g-ug_,). \end{array}$

Example 1 Static Schwartzschild space.

1.3. The positve mass theorem

Theorem 1 (Schoen-Yau 1979) If scalar curvature is ${\geq 0}$, then ${m(g)\geq 0}$, equalty implies that manifold is Euclidean.

In Newtonian gravity, ${m(g)}$ is the integral of mass density, so it is positive. But in general relativity, ${m}$ depends non linearly on the metric, making our life harder.

1.4. Idea of proof

One constructs an asymptotically plane minimal surface ${\Sigma}$ in ${M}$. For this, solve Plateau problems on larger and larger circles. Assuming ${m(g)<0}$ allows to give an upper bound of the height of ${\Sigma}$, and show convergence to a stable minimal surface.

In the second variation of area, one uses as test functions functions locally converging to 1 but with Dirichlet energy tending to 0. With Gauss-Bonnet, this yields negative second variation, contradiction.

1.5. Witten’s approach

On spin manifolds, the Dirac operator ${D}$ exists. It satisfies ${D=\nabla^{*}\nabla+\frac{1}{4}R}$. Integrating this by parts, there is a boundary term involving mass, yielding an expression for mass, which is obviously positive if ${R\geq 0}$.

1.6. Conformal blow ups

This is a special class of asymptotically flat manifolds, obtaining by conformally changing an arbitrary metric by the square of the Green function of the conformal Laplacian.

2. CR manifolds

2.1. Asymptotic flatness

Say a CR manifold has positive Webster class if there is a conformal metric with positive Webster curvature. Equivalently, the conformal Laplacian has positive first eigenvalue. Then Green’s function is well defined. Jerison-Lee 1989 give an asymptotic expansion that shows that the conformal metric ${Gg}$ is asymptotically flat (i.e., Heisenberg).

Definition 2 A pseudohermitian manifold is asymptotically flat if it is Heisenberg group equipped with a metric asymptotic to left-inavriant metric, and Webster curvature is in ${L^1}$.

2.2. Mass

The same logic (first variation of total Webster curvature) leads to the following

Definition 3

$\displaystyle \begin{array}{rcl} m(J,\theta)=\int_{S_r}\omega_1^1 \end{array}$

2.3. The Paneitz operator

It is conformally covariant third order operator arising in the characterization of CR pluri-harmonic functions (Lee 1988) and Szegö kernel expansion (Hirachi 1993).

Theorem 4 (Chanillo, Chiu, Yang 2010) If ${P\geq 0}$ and ${m>0}$,…

2.4. Integral formula for the mass

On an asymptotically flat pseudohermitian manifold, let ${\beta}$ be a function asymptotic to ${\bar{z}}$. Then mass is expressed as an integral of derivatives of ${\beta}$. If ${\beta}$ solves a certain differential equation, only one term is negative, it vanishes if box${(\beta)=0}$. One manages to find a solution ${\beta}$ of these equations, first up to a small error, then applying

2.5. Our result

Theorem 5 ${M}$ compact CR 3-manifold. Assume Webster class is positive, and Paneitz operator is nonnegative. Blow up conformally using Green’s function. Then

1. Resulting manifold is asymptotically flat.
2. Mass is ${\geq 0}$.
3. ${m=0}$ implies ${M}$ is CR equivalent to the round CR sphere.

2.6. Application to the CR Yamabe problem

This amounts to finding conformal metrics with constant Webster curvature. This can be done by minimizing Yamabe’s functional: Total scalar curvature normalized by volume. Loss of compactness is possible (due to conformal invariance). The infimum is always bounded above by the Yamabe invariant of the round CR sphere. If inequality is strict, then compactness is recovered (Jerison-Lee 1989).

These authors prove strict inequality if dimension is ${\geq 5}$ and metric non locally conformally flat, by a Taylor expansion. In low dimension, positive mass implies success of a similar procedure.

2.7. Example with positive Webster class but negative mass

This shows the necessity of some condition like positivity of Paneitz operator.

For this, we perturb Heisenberg metric in such a way that ${E_{11}}$ is a CR function. Positivity of Paneitz operator fails in this example.

2.8. Open problems

Can one use minimal surfaces ? Area makes sense. Its second variation has been computed (Cheng, Hwang, Malchiodi, Yang 2005), it seems complicated.

Quantitative lower bounds on the mass ? Like Penrose’s inequality: ${m\geq \frac{A}{16\pi}}$, ${A}$ is area of deepest minimal surface.

Compactness of solutions (Brendle-Marques, Khuri-Marques-Schoen 2009, assuming positive mass).

Classification of profiles of bubbles.