Search

In Special Relativity, massless objects are characterized as either vacuum states or as radiation propagating at the speed of light. This distinction extends to General Relativity for asymptotically flat initial data sets (IDS) \((M^n, g, k)\), where vacuum is represented by slices of Minkowski space, and radiation is modeled by slices of \(pp\)-wave spacetimes. In contrast, we demonstrate that asymptotically hyperboloidal IDS with zero mass must embed isometrically into Minkowski space, with no possible IDS configurations modeling radiation in this setting. Our result holds under the most general assumptions. The proof relies on precise decay estimates for spinors on level sets of spacetime harmonic functions and works in all dimensions., Comment: 23 pages; comments welcome!

We establish new mean curvature rigidity theorems of spin fill-ins with non-negative scalar curvature using two different spinorial techniques. Our results address two questions by Miao and Gromov, respectively. The first technique is based on extending boundary spinors satisfying a generalized eigenvalue equation via the Fredholm alternative for an APS boundary value problem, while the second is a comparison result in the spirit of Llarull and Lott using index theory. We also show that the latter implies a new Witten-type integral inequality for the mass of an asymptotically Schwarzschild manifold which holds even when the scalar curvature is not assumed to be non-negative., Comment: 22 pages; v2: additional result in section 3 and miscellaneous minor improvements

In 1981, Schoen-Yau and Witten showed that in General Relativity both the total energy $E$ and the total mass $m$ of an initial data set modelling an isolated gravitational system are non-negative. Moreover, if $E=0$, the initial data set must be contained in Minkowski space. In this paper, we show that if $m=0$, i.e. if $E$ equals the total momentum $|P|$, the initial data set must be contained in a pp-wave spacetime. Our proof combines spinorial methods with spacetime harmonic functions and works in all dimensions. Additionally, we find the decay rate threshold where the embedding has to be within Minkowski space and construct non-vacuum initial data sets with $m=0$ in the borderline case. As a consequence, this completely settles the rigidity of the spacetime positive mass theorem for spin manifolds.

The Circle of Willis (CoW) is an important network of arteries connecting major circulations of the brain. Its vascular architecture is believed to affect the risk, severity, and clinical outcome of serious neuro-vascular diseases. However, characterizing the highly variable CoW anatomy is still a manual and time-consuming expert task. The CoW is usually imaged by two angiographic imaging modalities, magnetic resonance angiography (MRA) and computed tomography angiography (CTA), but there exist limited public datasets with annotations on CoW anatomy, especially for CTA. Therefore we organized the TopCoW Challenge in 2023 with the release of an annotated CoW dataset. The TopCoW dataset was the first public dataset with voxel-level annotations for thirteen possible CoW vessel components, enabled by virtual-reality (VR) technology. It was also the first large dataset with paired MRA and CTA from the same patients. TopCoW challenge formalized the CoW characterization problem as a multiclass anatomical segmentation task with an emphasis on topological metrics. We invited submissions worldwide for the CoW segmentation task, which attracted over 140 registered participants from four continents. The top performing teams managed to segment many CoW components to Dice scores around 90%, but with lower scores for communicating arteries and rare variants. There were also topological mistakes for predictions with high Dice scores. Additional topological analysis revealed further areas for improvement in detecting certain CoW components and matching CoW variant topology accurately. TopCoW represented a first attempt at benchmarking the CoW anatomical segmentation task for MRA and CTA, both morphologically and topologically., Comment: 24 pages, 11 figures, 9 tables. Summary Paper for the MICCAI TopCoW 2023 Challenge

Llarull's theorem characterizes the round sphere $S^n$ among all spin manifolds whose scalar curvature is bounded from below by $n(n-1)$. In this paper we show that if the scalar curvature is bounded from below by $n(n-1)-\varepsilon$, the underlying manifold is $C^0$-close to a finite number of spheres outside a small bad set. This completely solves Gromov's spherical stability problem.

We prove uniqueness of tangent cones for forced mean curvature flow, at both closed self-shrinkers and round cylindrical self-shrinkers, in any codimension. The corresponding results for mean curvature flow in Euclidean space were proven by Schulze and Colding-Minicozzi respectively. We adapt their methods to handle the presence of the forcing term, which vanishes in the blow-up limit but complicates the analysis along the rescaled flow. Our results naturally include the case of mean curvature flows in Riemannian manifolds., Comment: 27 pages, comments welcome

Generalized torical band inequalities give precise upper bounds for the width of compact manifolds with boundary in terms of positive pointwise lower bounds for scalar curvature, assuming certain topological conditions. We extend several incarnations of these results in which pointwise scalar curvature bounds are replaced with spectral scalar curvature bounds. More precisely, we prove upper bounds for the width in terms of the principal eigenvalue of the operator $-\Delta +cR$, where $R$ denotes scalar curvature and $c>0$ is a constant. Three separate strategies are employed to obtain distinct results holding in different dimensions and under varying hypotheses, namely we utilize spacetime harmonic functions, $\mu$-bubbles, and spinorial Callias operators. In dimension 3, where the strongest result is produced, we are also able to treat open and incomplete manifolds, and establish the appropriate rigidity statements. Additionally, a version of such spectral torus band inequalities is given where tori are replaced with cubes. Finally, as a corollary we generalize classical work of Schoen and Yau, on the existence of black holes due to concentration of matter, to higher dimensions and with alternate measurements of size., Comment: 25 pages, improved exposition

We derive local and global monotonic quantities associated to $p$-harmonic functions on manifolds with nonnegative scalar curvature. As applications, we obtain inequalities relating the mass of asymptotically flat $3$-manifolds, the $p$-capacity and the Willmore functional of the boundary. As $ p \to 1$, one of the results retrieves a classic relation that the ADM mass dominates the Hawking mass if the surface is area outer-minimizing., Comment: 38 pages, to appear in PAMQ

We introduce new systems of PDE on initial data sets $(M,g,k)$ whose solutions model double-null foliations. This allows us to generalize Geroch's monotonicity formula for the Hawking mass under inverse mean curvature flow to initial data sets satisfying the dominant energy condition. We study the existence theory of these systems and give geometric applications., Comment: Simplified proofs, improved exposition

Comparison theorems are foundational to our understanding of the geometric features implied by various curvature constraints. This paper considers manifolds with a positive lower bound on either scalar, 2-Ricci, or Ricci curvature, and contains a variety of theorems which provide sharp relationships between this bound and notions of {\em{width}}. Some inequalities leverage geometric quantities such as boundary mean curvature, while others involve topological conditions in the form of linking requirements or homological constraints. In several of these results open and incomplete manifolds are studied, one of which partially addresses a conjecture of Gromov in this setting. The majority of results are accompanied by rigidity statements which isolate various model geometries -- both complete and incomplete -- including a new characterization of round lens spaces, and other models that have not appeared elsewhere. As a byproduct, we additionally give new and quantitative proofs of several classical comparison statements such as Bonnet-Myers' and Frankel's Theorem, as well as a version of Llarull's Theorem and a notable fact concerning asymptotically flat manifolds. The results that we present vary significantly in character, however a common theme is present in that the lead role in each proof is played by \emph{spacetime harmonic functions}, which are solutions to a certain elliptic equation originally designed to study mass in mathematical general relativity., Comment: Minor technical improvements

The Theorem of Bonnet--Myers implies that manifolds with topology $M^{n-1} \times \mathbb{S}^1$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus $\mathbb{T}^n$ does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so called $m$-intermediate curvature), and use stable weighted slicings to show that for $n \leq 7$ the manifolds $N^n = M^{n-m} \times \mathbb{T}^m$ do not admit a metric of positive $m$-intermediate curvature., Comment: final version; to appear in Comm. Pure. Appl. Math

The rigidity of the spacetime positive mass theorem states that an initial data set $(M,g,k)$ satisfying the dominant energy condition with vanishing mass can be isometrically embedded into Minkowski space. This has been established by Beig-Chru\'sciel and Huang-Lee under additional decay assumptions for the energy and momentum densities $\mu$ and $J$. In this note we give a new and elementary proof in dimension 3 which removes these additional decay assumptions. Our argument uses spacetime harmonic functions and Liouville's theorem. We also provide an alternative proof based on the Killing development of $(M,g,k)$., Comment: Comments welcome!

Effective visualizations were evaluated to reveal relevant health patterns from multi-sensor real-time wearable devices that recorded vital signs from patients admitted to hospital with COVID-19. Furthermore, specific challenges associated with wearable health data visualizations, such as fluctuating data quality resulting from compliance problems, time needed to charge the device and technical problems are described. As a primary use case, we examined the detection and communication of relevant health patterns visible in the vital signs acquired by the technology. Customized heat maps and bar charts were used to specifically highlight medically relevant patterns in vital signs. A survey of two medical doctors, one clinical project manager and seven health data science researchers was conducted to evaluate the visualization methods. From a dataset of 84 hospitalized COVID-19 patients, we extracted one typical COVID-19 patient history and based on the visualizations showcased the health history of two noteworthy patients. The visualizations were shown to be effective, simple and intuitive in deducing the health status of patients. For clinical staff who are time-constrained and responsible for numerous patients, such visualization methods can be an effective tool to enable continuous acquisition and monitoring of patients' health statuses even remotely., Comment: 17 pages, 9 figures, conference

In the pioneering work of Stern, level sets of harmonic functions have been shown to be an effective tool in the study of scalar curvature in dimension 3. Generalizations of this idea, utilizing level sets of so called spacetime harmonic functions as well as other elliptic equations, are similarly effective in treating geometric inequalities involving the ADM mass. In this paper, we survey recent results in this context, focusing on applications of spacetime harmonic functions to the asymptotically flat and asymptotically hyperbolic versions of the spacetime positive mass theorem, and additionally introduce a new concept of total mass valid in both settings which is encoded in interpolation regions between generic initial data and model geometries. Furthermore, a novel and elementary proof of the positive mass theorem with charge is presented, and the level set approach to the Penrose inequality given by Huisken and Ilmanen is related to the current developments. Lastly, we discuss several open problems., Comment: 27 pages, 3 figures. Contribution to the volume "Perspectives in Scalar Curvature"

We study the mass of asymptotically flat $3$-manifolds with boundary using the method of Bray-Kazaras-Khuri-Stern. More precisely, we derive a mass formula on the union of an asymptotically flat manifold and fill-ins of its boundary, and give new sufficient conditions guaranteeing the positivity of the mass. Motivation to such consideration comes from studying the quasi-local mass of the boundary surface. If the boundary isometrically embeds in the Euclidean space, we apply the formula to obtain convergence of the Brown-York mass along large surfaces tending to $\infty$ which include the scaling of any fixed coordinate-convex surface., Comment: 30 pages, 2 figures

We give a lower bound for the Lorentz length of the ADM energy-momentum vector (ADM mass) of 3-dimensional asymptotically flat initial data sets for the Einstein equations. The bound is given in terms of linear growth `spacetime harmonic functions' in addition to the energy-momentum density of matter fields, and is valid regardless of whether the dominant energy condition holds or whether the data possess a boundary. A corollary of this result is a new proof of the spacetime positive mass theorem for complete initial data or those with weakly trapped surface boundary, and includes the rigidity statement which asserts that the mass vanishes if and only if the data arise from Minkowski space. The proof has some analogy with both the Witten spinorial approach as well as the marginally outer trapped surface (MOTS) method of Eichmair, Huang, Lee, and Schoen. Furthermore, this paper generalizes the harmonic level set technique used in the Riemannian case by Bray, Stern, and the second and third authors, albeit with a different class of level sets. Thus, even in the time-symmetric (Riemannian) case a new inequality is achieved., Comment: J. Differential Geom., to appear

We consider the mean curvature flow of compact convex surfaces in Euclidean $3$-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on the geometry of the barrier, the flow contracts the surface to a point in finite time. Moreover, the solution is asymptotic to a shrinking half-sphere lying in a half space. This extends, in dimension two, the convergence result of Stahl for umbilic barriers to general convex barriers. We introduce a new perturbation argument to establish fundamental convexity and pinching estimates for the flow. Our result can be compared to a celebrated convergence theorem of Huisken for mean curvature flow of convex hypersurfaces in Riemannian manifolds., Comment: 32 pages, 1 figure

Let $X$ be a closed $3$-manifold, $\mathcal{M}_{R>0}$ the space of metrics on $X$ with positive scalar curvature, and $\text{Diff}(X)$ the group of diffeomorphisms of $X$. Marques proves the fundamental result that $\mathcal{M}_{R>0}/ \text{Diff}(X)$ is path connected. Using this and the theorem of Cerf in differential topology, Marques shows that the space of asymptotically flat metrics with nonnegative scalar curvature on $\mathbb{R}^3$ is path connected. Based on Carlotto-Li's generalization of Marques' result to the case of compact manifold with boundary, we show that the space of asymptotically flat metrics with nonnegative scalar curvature and mean convex boundary on $\mathbb{R}^3\backslash B^3$ is path connected. The differential topology part of Marques' argument no longer yields the desired result for $\mathbb{R}^3\setminus B^3$, but we bypass this issue by finding a more elementary proof. We also include a path-connectedness result for the space of black hole initial data sets, which can be thought of as a necessary condition for the Final State Conjecture., Comment: 14 pages, 1 figure

Flatly Foliated Relativity (FFR) is a new theory which conceptually lies between Special Relativity (SR) and General Relativity (GR), in which spacetime is foliated by flat Euclidean spaces. While GR is based on the idea that "matter curves spacetime", FFR is based on the idea that "matter curves spacetime, but not space". This idea, inspired by the observed spatial flatness of our local universe, is realized by considering the same action as used in GR, but restricting it only to metrics which are foliated by flat spatial slices. FFR can be thought of as describing gravity without gravitational waves. In FFR, a positive cosmological constant implies several interesting properties which do not follow in GR: the metric equations are elliptic on each euclidean slice, there exists a unique vacuum solution among those spherically symmetric at infinity, and there exists a geometric way to define the arrow of time. Furthermore, as gravitational waves do not exist in FFR, there are simple analogs to the positive mass theorem and Penrose-type inequalities. Importantly, given that gravitational waves have a negligible effect on the curvature of spacetime, and that the universe appears to be locally flat, FFR may be a good approximation of GR. Moreover, FFR still admits many notable features of GR including the big bang, an accelerating expansion of the universe, and the Schwarzschild spacetime. Lastly, FFR is already known to have an existence theory for some simplified cases, which provokes an interesting discussion regarding the possibility of a more general existence theory, which may be relevant to understanding existence of solutions to GR., Comment: 29 pages, 2 figures

We derive a positive mass theorem for asymptotically flat manifolds with boundary whose mean curvature satisfies a sharp estimate involving the conformal Green's function. The theorem also holds if the conformal Green's function is replaced by the standard Green's function for the Laplacian operator. As an application, we obtain an inequality relating the mass and harmonic functions that generalizes H. Bray's mass-capacity inequality in his proof of the Riemannian Penrose conjecture., Comment: 15 pages, to appear in Pacific J. Math