scholarly journals Capacity, quasi-local mass, and singular fill-ins

2020 ◽  
Vol 2020 (768) ◽  
pp. 55-92 ◽  
Author(s):  
Christos Mantoulidis ◽  
Pengzi Miao ◽  
Luen-Fai Tam
Keyword(s):  
Scalar Curvature ◽  
The Other ◽  
Independent Interest ◽  
Asymptotically Flat ◽  
Local Mass ◽  
Singular Metrics ◽  
Comparison Results ◽  
Mean Convex ◽  
New Inequalities

AbstractWe derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown–York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fill-ins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them.

scholarly journals Jordan products of quantum channels and their compatibility

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Mark Girard ◽  
Martin Plávala ◽  
Jamie Sikora
Keyword(s):  
Quantum State ◽  
Sufficient Conditions ◽  
The Other ◽  
Independent Interest ◽  
Quantum Channels ◽  
Semidefinite Programs ◽  
Jordan Product ◽  
Marginal Problem ◽  
Jordan Products ◽  
Product Of Matrices

AbstractGiven two quantum channels, we examine the task of determining whether they are compatible—meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolarizing channels are compatible.

scholarly journals Isoperimetry, Scalar Curvature, and Mass in Asymptotically Flat Riemannian 3‐Manifolds

2021 ◽  
Vol 74 (4) ◽  
pp. 865-905
Author(s):  
Otis Chodosh ◽  
Michael Eichmair ◽  
Yuguang Shi ◽  
Haobin Yu
Keyword(s):  
Scalar Curvature ◽  
Asymptotically Flat

scholarly journals On far-outlying constant mean curvature spheres in asymptotically flat Riemannian 3-manifolds

2020 ◽  
Vol 2020 (767) ◽  
pp. 161-191
Author(s):  
Otis Chodosh ◽  
Michael Eichmair
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Existence Results ◽  
Asymptotically Flat ◽  
Weingarten Surfaces ◽  
Differential Geom

AbstractWe extend the Lyapunov–Schmidt analysis of outlying stable constant mean curvature spheres in the work of S. Brendle and the second-named author [S. Brendle and M. Eichmair, Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94 2013, 3, 387–407] to the “far-off-center” regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable constant mean curvature spheres that depend delicately on the behavior of scalar curvature at infinity.

Discs area-minimizing in mean convex Riemannian n-manifolds

2021 ◽  
pp. 1-22
Author(s):  
Ezequiel Barbosa ◽  
Franciele Conrado
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Higher Dimensions ◽  
Dimensional Torus ◽  
Rigidity Result ◽  
Equality Case ◽  
Totally Geodesic ◽  
Zero Degree ◽  
Area Minimizing ◽  
Mean Convex

In this work, we consider oriented compact manifolds which possess convex mean curvature boundary, positive scalar curvature and admit a map to $\mathbb {D}^{2}\times T^{n}$ with non-zero degree, where $\mathbb {D}^{2}$ is a disc and $T^{n}$ is an $n$ -dimensional torus. We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed discs whose boundaries are homotopically non-trivial curves. We also prove a rigidity result for the equality case when the boundary is strongly totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in (2015, J. Geom. Anal., 25, 1001–1017) to higher dimensions.

scholarly journals Two examples related to conical energies

2022 ◽  
Vol 47 (1) ◽  
pp. 261-281
Author(s):  
Damian Dąbrowski
Keyword(s):  
Singular Integral ◽  
Recent Article ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
The Other ◽  
Independent Interest ◽  
Sufficient Condition ◽  
Lipschitz Graphs ◽  
Big Pieces

In a recent article (2021) we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and Mörters (2000), the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.

scholarly journals A basic inequality for submanifolds in a cosymplectic space form

2003 ◽  
Vol 2003 (9) ◽  
pp. 539-547 ◽  
Author(s):  
Jeong-Sik Kim ◽  
Jaedong Choi
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Mean Curvature ◽  
Space Form ◽  
The Other ◽  
Space Forms ◽  
Basic Inequality

For submanifolds tangent to the structure vector field in cosymplectic space forms, we establish a basic inequality between the main intrinsic invariants of the submanifold, namely, its sectional curvature and scalar curvature on one side; and its main extrinsic invariant, namely, squared mean curvature on the other side. Some applications, including inequalities between the intrinsic invariantδMand the squared mean curvature, are given. The equality cases are also discussed.

scholarly journals Total Energy of Charged Black Holes in Einstein-Maxwell-Dilaton-Axion Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Murat Korunur ◽  
Irfan Açıkgöz
Keyword(s):  
Black Holes ◽  
Total Energy ◽  
Energy Content ◽  
Teleparallel Gravity ◽  
The Other ◽  
Energy Distributions ◽  
Asymptotically Flat ◽  
Charged Black Holes ◽  
Momentum Complex ◽  
Limiting Case

We focus on the energy content (including matter and fields) of the Møller energy-momentum complex in the framework of Einstein-Maxwell-Dilaton-Axion (EMDA) theory using teleparallel gravity. We perform the required calculations for some specific charged black hole models, and we find that total energy distributions associated with asymptotically flat black holes are proportional to the gravitational mass. On the other hand, we see that the energy of the asymptotically nonflat black holes diverge in a limiting case.

scholarly journals Deformation of finite morphisms and smoothing of ropes

2008 ◽  
Vol 144 (3) ◽  
pp. 673-688 ◽  
Author(s):  
Francisco Javier Gallego ◽  
Miguel González ◽  
Bangere P. Purnaprajna
Keyword(s):  
General Theory ◽  
Projective Space ◽  
General Result ◽  
The Other ◽  
Independent Interest ◽  
Higher Dimension ◽  
Smooth Curves ◽  
Other Hand ◽  
Finite Covers ◽  
The One

AbstractIn this paper we prove that most ropes of arbitrary multiplicity supported on smooth curves can be smoothed. By a rope being smoothable we mean that the rope is the flat limit of a family of smooth, irreducible curves. To construct a smoothing, we connect, on the one hand, deformations of a finite morphism to projective space and, on the other hand, morphisms from a rope to projective space. We also prove a general result of independent interest, namely that finite covers onto smooth irreducible curves embedded in projective space can be deformed to a family of 1:1 maps. We apply our general theory to prove the smoothing of ropes of multiplicity 3 on P1. Even though this paper focuses on ropes of dimension 1, our method yields a general approach to deal with the smoothing of ropes of higher dimension.

RESULTS RELATED TO PRESCRIBING PSEUDO-HERMITIAN SCALAR CURVATURE

2013 ◽  
Vol 24 (03) ◽  
pp. 1350020 ◽  
Author(s):  
PAK TUNG HO
Keyword(s):  
Scalar Curvature ◽  
Smooth Function ◽  
The Other ◽  
Cr Manifold ◽  
Geometric Flow ◽  
Yamabe Invariant ◽  
Uniqueness Results ◽  
Other Hand

In this paper, we consider the problem of prescribing pseudo-Hermitian scalar curvature on a compact strictly pseudoconvex CR manifold M. Using geometric flow, we prove that for any negative smooth function f we can prescribe the pseudo-Hermitian scalar curvature to be f, provided that dim M = 3 and the CR Yamabe invariant of M is negative. On the other hand, we establish some uniqueness and non-uniqueness results on prescribing pseudo-Hermitian scalar curvature.

CONFORMAL COUPLING'S OUTRAGEOUS LEGACY

2002 ◽  
Vol 11 (10) ◽  
pp. 1531-1536
Author(s):  
L. RAUL ABRAMO ◽  
LEON BRENIG ◽  
EDGARD GUNZIG
Keyword(s):  
Scalar Field ◽  
Scalar Curvature ◽  
The Other ◽  
Minimal Coupling ◽  
Quantum Stability ◽  
Paradoxical Situation ◽  
Other Hand ◽  
The One ◽  
Hilbert Action ◽  
Scalar Gravity

In Einstein's gravity, non-minimal coupling of a scalar field to the scalar curvature leads to a paradoxical situation. On the one hand, it opens the way to qualitatively new cosmological dynamics. On the other hand, there are sectors of non-minimally coupled scalar-gravity theories for which the Einstein–Hilbert action reverses its sign, which seems to indicate that the whole system is unstable. We show how conformal coupling bypasses this problem. Due to a subtle interplay between gravity and the scalar field, classical and quantum stability are guaranteed globally. This liberates conformal coupling from a serious obstacle. Inflationary solutions in the new sector are also presented, which are validated by current observations.

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