Three Quasi-Local Masses
We propose a definition of quasi-local mass based on the Penrose Inequality. Two further definitions are given by measuring distortions of the exponential map.
gr-qc↗
arXiv subjects
Marcus A. Khuri
Publications and source records attributed to Marcus A. Khuri.
Boundary Value Problems for Mixed Type Equations and Applications
In this paper we outline a general method for finding well-posed boundary value problems for linear equations of mixed elliptic and hyperbolic type, which extends previous techniques of Berezanskii, Didenko, and Friedrichs. This method is then used to study a particular class of fully nonlinear mixed type equations which arise in applications to differential geometry.
math.AP↗
Uniqueness of static vacuum Einstein metrics and the Bartnik quasi-local mass
We analyse the issue of uniqueness of solutions of the static vacuum Einstein equations with prescribed geometric or Bartnik boundary data. Large classes of examples are constructed where uniqueness fails. We then discuss the implications of this behavior for the Bartnik quasi-local mass. A variational characterization of Bartnik boundary data is also given.
gr-qc↗
On the Local Solvability of Darboux's Equation
We reduce the question of local nonsolvability of the Darboux equation, and hence of the isometric embedding problem for surfaces, to the local nonsolvability of a simple linear equation whose type is explicitly determined by the Gaussian curvature.
math.AP↗
Counterexamples to the Local Solvability of Monge-Ampere Equations in the Plane
In this paper, we present smooth examples of degenerate hyperbolic and mixed type Monge-Ampere equations in the plane, which do not admit a local C^3 solution.
math.AP↗
Local Solvability of a Class of Degenerate Monge-Ampere Equations and Applications to Geometry
We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These are: the problem of locally prescribed Gaussian curvature for surfaces in R^3, and the local isometric embedding problem for two-dimensional Riemannian manifolds. We prove a general local existence result for a large class of Monge-Ampere equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature possesses a nondegenerate critical point.
math.AP↗
The Local Isometric Embedding in R^3 of Two-Dimensional Riemannian Manifolds With Gaussian Curvature Changing Sign to Finite Order on a Curve
We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in R^3. We prove a general local existence result for a large class of Monge-Ampere equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature vanishes to arbitrary finite order on a single smooth curve.
math.AP↗
The Hoop Conjecture in Spherically Symmetric Spacetimes
We give general sufficient conditions for the existence of trapped surfaces due to concentration of matter in spherically symmetric initial data sets satisfying the dominant energy condition. These results are novel in that they apply and are meaningful for arbitrary spacelike slices, that is they do not require any auxiliary assumptions such as maximality, time-symmetry, or special extrinsic foliations, and most importantly they can easily be generalized to the nonspherical case once an existence theory for a modified version of the Jang equation is developed. Moreover, our methods also yield positivity and monotonicity properties of the Misner-Sharp energy.
math.DG↗
Nonexistence of Generalized Apparent Horizons in Minkowski Space
We establish a Positive Mass Theorem for initial data sets of the Einstein equations having generalized trapped surface boundary. In particular we answer a question posed by R. Wald concerning the existence of generalized apparent horizons in Minkowski space.
math.DG↗
A Jang Equation Approach to the Penrose Inequality
We introduce a generalized version of the Jang equation, designed for the general case of the Penrose Inequality in the setting of an asymptotically flat space-like hypersurface of a spacetime satisfying the dominat energy condition. The appropriate existence and regularity results are established in the special case of spherically symmetric Cauchy data, and are applied to give a new proof of the general Penrose Inequality for these data sets. When appropriately coupled with an inverse mean curvature flow, analogous existence and regularity results for the associated system of equations in the nonspherical setting would yield a proof of the full Penrose Conjecture. Thus it remains as an important and challenging open problem to determine whether this system does indeed admit the desired solutions.
math.DG↗
P.d.e.'s which imply the Penrose conjecture
In this paper, we show how to reduce the Penrose conjecture to the known Riemannian Penrose inequality case whenever certain geometrically motivated systems of equations can be solved. Whether or not these special systems of equations have general existence theories is therefore an important open problem. The key tool in our method is the derivation of a new identity which we call the generalized Schoen-Yau identity, which is of independent interest. Using a generalized Jang equation, we propose canonical embeddings of Cauchy data into corresponding static spacetimes. In addition, our techniques suggest a more general Penrose conjecture and generalized notions of apparent horizons and trapped surfaces, which are also of independent interest.
math.DG↗