Nancy scalpotmas2026

Here are some references for my talks at the Nancy scalpotmas2026 Masterclass. 

Papers I will be discussing the content of:

• Harmonic Functions and The Mass of 3-Dimensional Asymptotically Flat Riemannian Manifolds

• Spacetime Harmonic Functions and the Mass of 3-Dimensional Asymptotically Flat Initial Data for the Einstein Equations 

• Rigid comparison geometry for Riemannian bands and open incomplete manifolds

• Stability of Euclidean 3-space for the positive mass theorem

Here are supplementary resources:

• Chru´sciel's notes on energy in General Relativity

• Bartnik's paper on the mass of asymptotically flat manifolds

• Gromov and Lawson's big IHES paper on positive scalar curvature manifolds

• Schoen and Yau's proof of the positive mass theorem

• Dan Lee's textbook on mathematical (Riemannian) General Relativity

• Daniel Stern's paper on scalar curvature of 3-manifolds and harmonic maps to S^1

• My paper with Allen-Bryden-Khuri on the suboptimal Penrose inequality 

Schedule of talks, subject to change:

Day 1: Introduction to scalar curvature, overview of minimal surface and Dirac operator methods, fundamental scalar curvature level set formula

Day 2: Introduction to mass in General Relativity, the dominant energy condition and constraint equations, sketch of proof of the Riemannian positive mass theorem by harmonic functions

Day 3: Spacetime harmonic functions, the spacetime positive mass theorem, application to band inequalities, application to Llarull's teorem

Day 4: Sketch of the C^0 Dong-Song mass stability result. Sketch of application to the suboptimal Penrose inequality.

Day 1 Exercises: 

1. [In the proof of the rigidity of the systole inequality using harmonic maps to S^1] Show that the Gauss curvatures of the level sets of \tilde{u} are constant. 

2. Prove that the second fundamental form of a regular level set is the tangential part of the Hessian divided by |\nabla u|.

3. Prove X(|\nabla u|)=Hess u (X,\nu)

4. Prove the decomposition formula for the Laplacian over a hypersurface S with unit normal \eta: \Delta u=\Delta^S u + H^S \eta(u) + \nabla_{\eta \eta} u.

5. Read Daniel Stern's article, particularly the part that deals with critical points of a harmonic map.

6. Show that, for a harmonic function u on a domain \Omega (and no critical points on \partial \Omega), that 

\int_{\partial \Omega} [ n(|\nabla u|)-|\nabla u|\kappa_t ]dA

is equal to 

-\int_{\partial \Omega} [ |\nabla ^T u|/|\nabla u| ] <\nabla ^T u ,\nabla (n(u)/|\nabla^T u|)>dA

where n is the outward unit normal and \nabla ^T u denotes the tangential part of \nabla u on \partial Omega.