In this paper, we prove compactness for the full set of solutions to Yamabe Problem if $n\leq 24$. After proving sharp pointwise estimates at a blowup point, Weyl Vanishing The- orem in those dimensions, and reduce question showing positivity quadratic form. We also show that form has negative eigenvalues 25$.

We give a lower bound for the Lorentz length of ADM energy-momentum vector (ADM mass) 3‑dimensional asymptotically flat initial data sets Einstein equations. The is given in terms linear growth 'spacetime harmonic functions' addition to density matter fields, and valid regardless whether dominant energy condition holds or possess boundary. A corollary this result new proof spacetime positive mass theorem complete those with weakly trapped surface boundary, includes rigidity statement which...

In this paper, we show how to reduce the Penrose conjecture known Riemannian inequality case whenever certain geometrically motivated systems of equations can be solved.Whether or not these special have general existence theories is therefore an important open problem.The key tool in our method derivation a new identity which call generalized Schoen-Yau identity, independent interest.Using Jang equation, use propose canonical embeddings Cauchy data into corresponding static spacetimes.In...

In [4] and [5], Bray Khuri outlined an approach to prove the Penrose inequality for general initial data sets of Einstein equations.In this paper we extend so that it may be applied a charged version inequality.Moreover, assuming is time symmetric, rigidity statement in case equality inequality, result which seems absent from literature.A new quasi-local mass, tailored also introduced, used proof.

We present a proof of the Riemannian Penrose inequality with charge in context asymptotically flat initial data sets for Einstein–Maxwell equations, having possibly multiple black holes no charged matter outside horizon, and satisfying relevant dominant energy condition. The is based on generalization Hubert Bray's conformal flow metrics adapted to this setting.

This is the second in a series of two papers to establish conjectured mass-angular momentum inequality for multiple black holes, modulo extreme hole 'no hair theorem'. More precisely it shown that either there counterexample uniqueness, form regular axisymmetric stationary vacuum spacetime with an asymptotically flat end and degenerate horizons which 'ADM minimizing', or following statement holds. Complete, simply connected, maximal initial data sets Einstein equations ends are cylindrical,...

A classic result of Shi and Tam states that a 2-sphere positive Gauss mean curvature bounding compact 3-manifold with nonnegative scalar must have total not greater than the isometric embedding into Euclidean 3-space, equality only for domains in this reference manifold. We generalize to 2-tori <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative 1"> <mml:semantics> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow>...

We introduce a generalized version of the Jang equation, designedfor general case Penrose Inequality in setting ofan asymptotically flat space-like hypersurface spacetimesatisfying dominant energy condition. The appropriateexistence and regularity results are established specialcase spherically symmetric Cauchy data, applied to givea new proof for these data sets.When appropriately coupled with an inverse mean curvature flow,analogous existence associatedsystem equations nonspherical would...

The most general formulation of Penrose's inequality yields a lower bound for ADM mass in terms the area, charge, and angular momentum black holes. This is turn equivalent to an upper area remaining quantities. In this note, we establish single hole setting axisymmetric maximal initial data sets Einstein-Maxwell equations, when non-electromagnetic matter fields are not charged satisfy dominant energy condition. It shown that saturated if only arise from extreme Kerr-Newman spacetime. Further...

We develop a framework for understanding the existence of asymptotically flat solutions to static vacuum Einstein equations on with geometric boundary conditions ∂M ≃ S2. A partial result is obtained, giving resolution conjecture Bartnik such extensions. The and uniqueness extensions closely related Bartnik's definition quasi-local mass.

Abstract The generalized Jang equation was introduced in an attempt to prove the Penrose inequality setting of general initial data for Einstein equations. In this paper we give extensive study equation, proving existence, regularity, and blow-up results. particular, precise asymptotics behavior are given, it is shown that solutions not unique. Keywords: Blow-up solutionsGeneralized equationPenrose inequalityMathematics Subject Classification: 35Q7535J7083C9958J32 Acknowledgments first...

In this paper, we show how a natural coupling of the Dirac equation with generalized Jang leads to proof rigidity statement in positive mass theorem charge, without maximal slicing condition, provided solution coupled system exists.

Abstract We study the problem of conformal deformation Riemannian structure to constant scalar curvature with zero mean on boundary. prove compactness for full set solutions when boundary is umbilic and dimension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mn>24</m:mn> </m:mrow> </m:math> ${n\leq 24}$ . The Weyl Vanishing Theorem also established under these hypotheses, we provide counter-examples <m:mo>≥</m:mo> <m:mn>25</m:mn> ${n\geq 25}$...

Abstract Generalized torical band inequalities give precise upper bounds for the width of compact manifolds with boundary in terms positive pointwise lower scalar curvature, assuming certain topological conditions. We extend several incarnations these results which curvature are replaced spectral bounds. More precisely, we prove principal eigenvalue operator $-\Delta +cR$, where $R$ denotes and $c&amp;gt;0$ is a constant. Three separate strategies employed to obtain distinct holding...

We give general sufficient conditions for the existence of trapped surfaces due to concentration matter in spherically symmetric initial data sets satisfying dominant energy condition. These results are novel that they apply and meaningful arbitrary spacelike slices, is, do not require any auxiliary assumptions such as maximality, time symmetry, or special extrinsic foliations, most importantly can easily be generalized nonspherical case once an theory a modified version Jang equation is...

We study the old problem of isometrically embedding a twodimensional Riemannian manifold into Euclidean three-space.It is shown that if Gaussian curvature vanishes to finite order and its zero set consists two Lipschitz curves intersecting transversely at point, then local sufficiently smooth isometric embeddings exist.

We study the problem of asymptotically flat bi-axially symmetric stationary solutions vacuum Einstein equations in 5-dimensional spacetime. In this setting, cross section any connected component event horizon is a prime 3-manifold positive Yamabe type, namely 3-sphere S3, ring S1×S2, or lens space L(p, q). The reduce to an axially harmonic map with prescribed singularities from R3 into SL(3,R)/SO(3). paper, we solve for all possible topologies, and particular first candidates smooth...