Abstract:
Consider an asymptotically flat Riemannian manifold $(M,g)$ of dimension $n \geq 3$ with nonempty compact boundary. We recall the harmonic conformal class $[g]_h$ of the metric, which consists of all conformal rescalings given by a harmonic function raised to an appropriate power. The geometric significance is that every metric in $[g]_h$ has the same pointwise sign of scalar curvature. For this reason, the harmonic conformal class appears in the study of general relativity, where scalar curvature is related to energy density. Our purpose is to introduce and study invariants of the harmonic conformal class. These invariants are closely related to constrained geometric optimization problems involving hypersurface area-minimizers and the ADM mass. In the final section, we discuss possible applications of the invariants and their relationship with zero area singularities and the positive mass theorem.

Abstract:
We recall the Riemannian Penrose inequality, a geometric statement that restricts the asymptotics of manifolds of nonnegative scalar curvature that contain compact minimal hypersurfaces. The inequality is known for manifolds of dimension up to seven. One source of motivation for the present work is to prove a weakened inequality in all dimensions for conformally flat manifolds. Along the way, we establish new inequalities, including some that apply to manifolds that are merely conformal to a particular type of scalar flat manifold (not necessarily conformally flat). We also apply the techniques to asymptotically flat manifolds containing zero area singularities, objects that generalize the naked singularity of the negative mass Schwarzschild metric. In particular we derive a lower bound for the ADM mass of a conformally flat, asymptotically flat manifold containing any number of zero area singularities.

Abstract:
The ADM mass, viewed as a functional on the space of asymptotically flat Riemannian metrics of nonnegative scalar curvature, fails to be continuous for many natural topologies. In this paper we prove that lower semicontinuity holds in natural settings: first, for pointed Cheeger--Gromov convergence (without any symmetry assumptions) for $n=3$, and second, assuming rotational symmetry, for weak convergence of the associated canonical embeddings into Euclidean space, for $n \geq 3$. We also apply recent results of LeFloch and Sormani to deal with the rotationally symmetric case, with respect to a pointed type of intrinsic flat convergence. We provide several examples, one of which demonstrates that the positive mass theorem is implied by a statement of the lower semicontinuity of the ADM mass.

Abstract:
Consider a triple of "Bartnik data" $(\Sigma, \gamma,H)$, where $\Sigma$ is a topological 2-sphere with Riemannian metric $\gamma$ and positive function $H$. We view Bartnik data as a boundary condition for the problem of finding a compact Riemannian 3-manifold $(\Omega,g)$ of nonnegative scalar curvature whose boundary is isometric to $(\Sigma,\gamma)$ with mean curvature $H$. Considering the perturbed data $(\Sigma, \gamma, \lambda H)$ for a positive real parameter $\lambda$, we find that such a "fill-in" $(\Omega,g)$ must exist for $\lambda$ small and cannot exist for $\lambda$ large; moreover, we prove there exists an intermediate threshold value. The main application is the construction of a new quasi-local mass, a concept of interest in general relativity. This mass has the nonnegativity property, but differs from many other definitions in that it tends to vanish on static vacuum (as opposed to flat) regions. We also recognize this mass as a special case of a type of twisted product of quasi-local mass functionals. Several ideas in this paper draw on work of Bray, Brendle--Marques--Neves, Corvino, Miao, and Shi--Tam.

Abstract:
We provide new results and new proofs of results about the torsion of curves in $\mathbb{R}^3$. Let $\gamma$ be a smooth curve in $\mathbb{R}^3$ that is the graph over a simple closed curve in $\mathbb{R}^2$ with positive curvature. We give a new proof that if $\gamma$ has nonnegative (or nonpositive) torsion, then $\gamma$ has zero torsion and hence lies in a plane. Additionally, we prove the new result that a simple closed plane curve, without any assumption on its curvature, cannot be perturbed to a closed space curve of constant nonzero torsion. We also prove similar statements for curves in Lorentzian $\mathbb{R}^{2,1}$ which are related to important open questions about time flat surfaces in spacetimes and mass in general relativity.

Abstract:
The Schwarzschild spacetime metric of negative mass is well-known to contain a naked singularity. In a spacelike slice, this singularity of the metric is characterized by the property that nearby surfaces have arbitrarily small area. We develop a theory of such "zero area singularities" in Riemannian manifolds, generalizing far beyond the Schwarzschild case (for example, allowing the singularities to have nontrivial topology). We also define the mass of such singularities. The main result of this paper is a lower bound on the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature in terms of the masses of its singularities, assuming a certain conjecture in conformal geometry. The proof relies on the Riemannian Penrose Inequality. Equality is attained in the inequality by the Schwarzschild metric of negative mass. An immediate corollary is a version of the Positive Mass Theorem that allows for certain types of incomplete metrics.

Abstract:
We identify a condition on spacelike 2-surfaces in a spacetime that is relevant to understanding the concept of mass in general relativity. We prove a formula for the variation of the spacetime Hawking mass under a uniformly area expanding flow and show that it is nonnegative for these so-called "time flat surfaces." Such flows generalize inverse mean curvature flow, which was used by Huisken and Ilmanen to prove the Riemannian Penrose inequality for one black hole. A flow of time flat surfaces may have connections to the problem in general relativity of bounding the mass of a spacetime from below by the quasi-local mass of a spacelike 2-surface contained therein.

Abstract:
In this sequel paper we give a shorter, second proof of the monotonicity of the Hawking mass for time flat surfaces under spacelike uniformly area expanding flows in spacetimes that satisfy the dominant energy condition. We also include a third proof which builds on a known formula and describe a class of sufficient conditions of divergence type for the monotonicity of the Hawking mass. These flows of surfaces may have connections to the problem in general relativity of bounding the total mass of a spacetime from below by the quasi-local mass of spacelike 2-surfaces in the spacetime.

Abstract:
In this paper we extend some well-known rigidity results for conformal changes of Einstein metrics to the class of generalized quasi-Einstein (GQE) metrics, which includes gradient Ricci solitons. In order to do so, we introduce the notions of conformal diffeomorphisms and vector fields that preserve a GQE structure. We show that a complete GQE metric admits a structure-preserving, non-homothetic complete conformal vector field if and only if it is a round sphere. We also classify the structure-preserving conformal diffeomorphisms. In the compact case, if a GQE metric admits a structure-preserving, non-homothetic conformal diffeomorphism, then the metric is conformal to the sphere, and isometric to the sphere in the case of a gradient Ricci soliton. In the complete case, the only structure-preserving non-homothetic conformal diffeomorphisms from a shrinking or steady gradient Ricci soliton to another soliton are the conformal transformations of spheres and stereographic projection.

Abstract:
Motivated by the quasi-local mass problem in general relativity, we apply the asymptotically flat extensions, constructed by Shi and Tam in the proof of the positivity of the Brown--York mass, to study a fill-in problem of realizing geometric data on a 2-sphere as the boundary of a compact 3-manifold of nonnegative scalar curvature. We characterize the relationship between two borderline cases: one in which the Shi--Tam extension has zero total mass, and another in which fill-ins of nonnegative scalar curvature fail to exist. Additionally, we prove a type of positive mass theorem in the latter case.