Table 4 Selection of ten open problems

Open Problem 2.13:

Conformal method as generic parametrisation of data sets

Context:

Conformal method

Status:

Partial results, for model problems

(Chruściel and Gicquaud 2017 and Maxwell 2011)

Open Problem 3.13:

Dynamical construction of exotic localised solutions

Context:

Characteristic data for the hyperbolic problem, gravitational radiation

Status:

Fully open

(no directly related contributions)

Open Problem 3.14:

Massive data that are flat in arbitrary domains

Context:

Gluing methods and Riemannian positive mass theorem

Status:

Fully open in the nonlinear case, unconstrained in the linear case

(Beig and Chruściel 2017)

Open Problem 3.16:

Novel counterexamples to the Min-Oo conjecture

Context:

Gluing methods and spherical positive mass theorem

Status:

Results with different techniques

(Brendle et al. 2011)

Open Problem 3.18:

Localised data with Newtonian decay

Context:

Gluing methods for asymptoptically flat data sets

Status:

Localisation modulo higher-order error terms

(LeFloch and Nguyen 2019)

Open Problem 3.21:

Scalar-flat collar extension of horizon data

Context:

Bartnik quasi-local mass (vacuum case)

Status:

Definitive results under DEC

(Mantoulidis and Schoen 2015)

Open Problem 3.23:

Existence of minimisers in the positive CMC case

Context:

Variational aspects of Bartnik quasi-local mass

Status:

Quantitative bounds but existence unclear

(Cabrera Pacheco et al. 2017)

Open Problem 3.26:

Effective Sobolev-type extension theorems for initial data

Context:

Extension and localised \(L^2\)-bounded conjecture

Status:

Perturbative results in the maximal case

(Czimek 2018)

Open Problem 4.4:

Constant scalar curvature metrics as differentiable manifold

Context:

Structure of the constraint manifold (\(\varLambda >0\))

Status:

Solved only for \(\varLambda =0\), (partial) generic regularity results

(Fischer and Marsden 1975a; Beig et al. 2005)

Open Problem 4.18:

Homotopy-type of the constraint manifold

Context:

Deformations of data and global shape

Status:

Path-connectedness in the maximal case

(Marques 2012; Carlotto and Li 2019)