An official website of the United States government
Abstract We study the prescribed scalar curvature problem in a conformal class on orbifolds with isolated singularities. We prove a compactness theorem in dimension 4, and an existence theorem which holds in dimensions$$n \ge 4$$ . This problem is more subtle than the manifold case since the positive mass theorem does not hold for ALE metrics in general. We also determine the$$\textrm{U}(2)$$ -invariant Leray–Schauder degree for a family of negative-mass orbifolds found by LeBrun.
more »
« less
This content will become publicly available on January 1, 2026
A survey on the positive mass theorem for asymptotically flat initial data sets
In honor of Yvonne Choquet-Bruhat’s 100th birthday, we present this survey on the positive mass theorem. Originating from a conjecture in general relativity regarding the ADM mass, the positive mass theorem has significantly influenced geometry and analysis over the past four decades and continued to inspire new connections. We review seminal contributions as well as recent advances, and then we focus our discussions on the equality case and the counter-examples arising from pp-wave spacetimes.
more »
« less
- Award ID(s):
- 1928930
- PAR ID:
- 10600024
- Publisher / Repository:
- Mersenne Center for open scientific publishing
- Date Published:
- Journal Name:
- Comptes Rendus. Mécanique
- Volume:
- 353
- Issue:
- G1
- ISSN:
- 1873-7234
- Page Range / eLocation ID:
- 177 to 194
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We construct asymptotically flat, scalar flat extensions of Bartnik data [Formula: see text], where [Formula: see text] is a metric of positive Gauss curvature on a two-sphere [Formula: see text], and [Formula: see text] is a function that is either positive or identically zero on [Formula: see text], such that the mass of the extension can be made arbitrarily close to the half area radius of [Formula: see text]. In the case of [Formula: see text], the result gives an analog of a theorem of Mantoulidis and Schoen [On the Bartnik mass of apparent horizons, Class. Quantum Grav. 32(20) (2015) 205002, 16 pp.], but with extensions that have vanishing scalar curvature. In the context of initial data sets in general relativity, the result produces asymptotically flat, time-symmetric, vacuum initial data with an apparent horizon [Formula: see text], for any metric [Formula: see text] with positive Gauss curvature, such that the mass of the initial data is arbitrarily close to the optimal value in the Riemannian Penrose inequality. The method we use is the Shi–Tam type metric construction from [Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62(1) (2002) 79–125] and a refined Shi–Tam monotonicity, found by the first named author in [On a localized Riemannian Penrose inequality, Commun. Math. Phys. 292(1) (2009) 271–284].more » « less
-
null (Ed.)Abstract We prove several rigidity results related to the spacetime positive mass theorem. A key step is to show that certain marginally outer trapped surfaces are weakly outermost. As a special case, our results include a rigidity result for Riemannian manifolds with a lower bound on their scalar curvature.more » « less
-
Abstract We study the fractional Yamabe problem first considered by Gonzalez-Qing [36] on the conformal infinity $$(M^{n}, \;[h])$$ of a Poincaré-Einstein manifold $$(X^{n+1}, \;g^{+})$$ with either $n=2$ or $$n\geq 3$$ and $$(M^{n}, \;[h])$$ locally flat, namely $(M, h),$ is locally conformally flat. However, as for the classical Yamabe problem, because of the involved quantization phenomena, the variational analysis of the fractional one exhibits a local situation and also a global one. The latter global situation includes the case of conformal infinities of Poincaré-Einstein manifolds of dimension either $n=2$ or of dimension $$n\geq 3$$ and which are locally flat, and hence the minimizing technique of Aubin [4] and Schoen [48] in that case clearly requires an analogue of the positive mass theorem of Schoen-Yau [49], which is not known to hold. Using the algebraic topological argument of Bahri-Coron [8], we bypass the latter positive mass issue and show that any conformal infinity of a Poincaré-Einstein manifold of dimension either $n=2$ or of dimension $$n\geq 3$$ and which is locally flat admits a Riemannian metric of constant fractional scalar curvature.more » « less
-
Vidick, Thomas (Ed.)Graph homomorphism has been an important research topic since its introduction [14]. Stated in the language of binary relational structures in that paper [14], Lovász proved a fundamental theorem that the graph homomorphism function G 7→ hom(G, H) for 0-1 valued H (as the adjacency matrix of a graph) determines the isomorphism type of H. In the past 50 years various extensions have been proved by Lovász and others [15, 9, 1, 19, 17]. These extend the basic 0-1 case to admit vertex and edge weights; but always with some restrictions such as all vertex weights must be positive. In this paper we prove a general form of this theorem where H can have arbitrary vertex and edge weights. An innovative aspect is that we prove this by a surprisingly simple and unified argument. This bypasses various technical obstacles and unifies and extends all previous known versions of this theorem on graphs. The constructive proof of our theorem can be used to make various complexity dichotomy theorems for graph homomorphism effective, i.e., it provides an algorithm that for any H either outputs a P-time algorithm solving hom(·, H) or a P-time reduction from a canonical #P-hard problem to hom(·, H).more » « less
