The study proves the regularity of inverse mean curvature flow in specific geometric settings.
problem Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds.
method Utilizing the behavior of Hawking masses, the study shows star-shaped slices after a long time.
result The weak solution of inverse mean curvature flow becomes regular over time.
On asymptotically flat and asymptotically hyperbolic manifolds, by evaluating the total mass via the Ricci tensor, we show that the limits of certain Brown-York type and Hawking type quasi-local mass integrals equal the total mass of the manifold in all dimensions.
It is well-know that Hawking mass is nonnegative for a stable constant mean curvature (CMC) sphere in three manifold of nonnegative scalar curvature. R. Bartnik proposed the rigidity problem of Hawking mass of stable CMC spheres. In this paper, we show partial rigidity results of Hawking mass for stable CMC spher…
In this paper, we will study the limiting behavior of the Brown-York mass of the coordinate spheres in an asymptotically flat manifold. Limiting behaviors of volumes of regions related to coordinate spheres are also obtained, including a discussion on the isoperimetric mass introduced by Huisken \cite{Huisken}. We will…
Sharp bounds for charged Hawking mass in electrostatic space-times.
problem Bounding charged Hawking mass in electrostatic space-times.
method Proving sharp lower bounds and upper bounds for the charged Hawking mass.
result Sharp lower bounds for the charged Hawking mass of stable surfaces in electrostatic space-times.
New PDE systems generalize Hawking mass monotonicity.
problem Generalizing Hawking mass monotonicity to initial data sets.
method Introduced new systems of PDE on initial data sets (M,g,k). result Generalized Geroch's monotonicity formula to initial data sets.
Two masses on surfaces with boundary converge to ADM mass.
problem Evaluating quasi-local masses on surfaces with boundaries.
method Hawking mass and Huisken's isoperimetric mass on surfaces with boundary, convergence to ADM mass.
result Convergence of Hawking and Huisken's masses to ADM mass.
In this paper, we study the relation of the monotonicity of Hawking Mass and geometric flow problems. We show that along the Hamilton-DeTurck flow with bounded curvature coupled with the modified mean curvature flow, the Hawking mass of the hypersphere with a sufficiently large radius in Schwarzschild spaces is monoton…
Rigidity results for Hawking mass in curved spaces with bounds on Bartnik capacity.
problem Rigidity of surfaces in curved spaces with mass bounds.
method Analyzing Hawking mass and applying rigidity results to specific geometric settings.
result Explicit lower bounds on Hawking and Bartnik masses in non-flat spaces.
The study examines symmetry of solutions to a mean field equation on the 2-sphere, leading to rigidity results for Hawking mass.
problem Symmetry of solutions to a specific elliptic equation on the 2-sphere.
method Analysis of the elliptic equation and conditions for constant solutions.
result The equation has only constant solutions under certain conditions, implying rigidity of Hawking mass.
Paper proves rigidity of 3-manifolds with boundary using modified Hawking mass.
problem Rigidity of 3-manifolds with boundary under specific geometric conditions.
method Area estimates for free boundary strictly stable two-disks, modified Hawking mass analysis.
result 3-manifolds with boundary are locally isometric to half anti-de Sitter-Schwarzschild manifold.
In this paper, we study the limiting behavior of the Brown-York mass and Hawking mass along nearly round surfaces at infinity of an asymptotically flat manifold. Nearly round surfaces can be defined in an intrinsic way. Our results show that the ADM mass of an asymptotically flat 3-manifold can be approximated by some …
In this sequel paper we give a shorter, second proof of the monotonicity of the Hawking mass for time flat surfaces under spacelike uniformly area expanding flows in spacetimes that satisfy the dominant energy condition. We also include a third proof which builds on a known formula and describe a class of sufficient co…
Paper proves rigidity of CMC surfaces in curved 3-manifolds.
problem Rigidity of CMC surfaces in positive curved 3-manifolds.
method Assumptions of surface being approximately round or invariant under even symmetry, and use of Hawking mass.
result Rigidity results for stable CMC surfaces with zero Hawking mass.
Given a constant mean curvature surface that bounds a compact manifold with nonnegative scalar curvature, we obtain intrinsic conditions on the surface that guarantee the positivity of its Hawking mass. We also obtain estimates of the Bartnik mass of such surfaces, without assumptions on the integral of the squared mea…
Proves Riemannian Penrose Inequality for specific manifolds.
problem Proving Riemannian Penrose Inequality for certain manifolds.
method Novel interplay between Hawking mass and potential-theoretic Hawking mass.
result Establishes equality between ADM mass and Huisken's Isoperimetric mass.
Study of area preserving Willmore flow on Schwarzschild-like manifolds.
problem Stability of Willmore spheres in Schwarzschild-like ends.
method Area preserving Willmore flow analysis in C3−close Schwarzschild ends. result Leaves of the Willmore foliation are strict local area preserving maximizers of Hawking mass.
In this paper, we prove that the even solution of the mean field equation Δu=λ(1−eu) on S2 must be axially symmetric when 4<λ≤8. In particular, zero is the only even solution for λ=6. This implies the rigidity of Hawking mass for stable constant mean curvature(CMC) sphere with even symmetry.
We study rigidity of minimal two-spheres Σ that locally maximize the Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature. After assuming strict stability of Σ, we prove that a neighborhood of it in M is isometric to one of the deSitter-Schwarzschild metrics on $(- ε,ε)\…
We identify a condition on spacelike 2-surfaces in a spacetime that is relevant to understanding the concept of mass in general relativity. We prove a formula for the variation of the spacetime Hawking mass under a uniformly area expanding flow and show that it is nonnegative for these so-called "time flat surfaces." S…
Study on stellar models' topology and mass using minimal surfaces.
problem Investigating the topology and mass of static stellar models.
method Analyzing stable free boundary minimal surfaces in static perfect fluid spaces.
result Proved non-existence of stable free boundary minimal surfaces and derived upper bounds for Hawking mass.
The paper explores stable surfaces in Einstein-Maxwell theory, proving mass bounds and nonexistence results.
problem Exploring stable surfaces in static Einstein-Maxwell space-time.
method Using mean-stable surfaces theory to prove properties of lapse functions and mass bounds.
result Proves ADM mass is bounded by Hawking quasi-local mass.
New mass-type invariants for cosmological space-times.
problem Characterize de Sitter solutions in space-times with a cosmological constant.
method Introduce new mass-type invariants and prove positive mass theorems.
result 1-harmonic Mass provides new characterizations and inequalities.
Upper bounds on Bartnik mass for non-negatively curved spheres.
problem Bounding Bartnik mass for non-negatively curved spheres.
method Establishing upper bounds using non-negative Gauss curvature.
result Upper bounds on Bartnik mass approach Hawking mass under certain conditions.
3D metrics get scalar curvature bounds via IMCF.
problem Bounding scalar curvature for C0 metrics. method Inverse Mean Curvature Flow (IMCF) and Hawking mass monotonicity.
result Stability theorem for nonnegative scalar curvature.
In this paper we characterize the intrinsic geometry of apparent horizons (outermost marginally outer trapped surfaces) in asymptotically flat spacetimes; that is, the Riemannian metrics on the two sphere which can arise. Furthermore we determine the minimal ADM mass of a spacetime containing such an apparent horizon. …
Study characterizes compact Einstein-type manifolds with boundary.
problem Characterize compact Einstein-type manifolds with nonempty boundary.
method Proved a sharp boundary estimate, obtained Hawking mass bounds, and provided a topological classification for the boundary.
result Obtained a gap result for compact Einstein-type manifolds with boundary.
Study rigidity of minimal disks in specific 3-manifolds.
problem Rigidity of free boundary minimal disks in mean convex three-manifolds.
method Assuming strict stability, prove isometric neighborhoods using modified Hawking mass.
result Prove rigidity of minimal disks in specific 3-manifolds.
Paper proves Penrose inequality with a weaker late-time condition.
problem Penrose's inequality under the black hole final state conjecture.
method Developed a new late-time condition called quasi final state hypothesis and proved the inequality.
result Proved the spacetime Penrose inequality under the quasi final state hypothesis.
In this paper we prove the Penrose inequality for metrics that are small perturbations of the Schwarzschild anti-de Sitter metrics of positive mass. We use the existence of a global foliation by weakly stable constant mean curvature spheres and the monotonicity of the Hawking mass.
In this paper, we obtain lower bounds for the Brown-York quasilocal mass and the Bartnik quasilocal mass for compact three manifolds with smooth boundaries. As a consequence, we derive sufficient conditions for the existence of horizons for a certain class of compact manifolds with boundary and some asymptotically flat…
Study on area-constrained Willmore spheres in asymptotic Schwarzschild manifolds.
problem Existence of area-constrained Willmore spheres with non-negative Hawking mass and inner radius.
method Analysis of scalar curvature and asymptotic properties of 3-manifolds.
result No large area-constrained Willmore spheres exist under certain conditions.
The Hawking energy is nonnegative and rigid on area-constrained surfaces in general relativity.
problem The rigidity and positivity of the Hawking energy on specific surfaces in general relativity.
method Evaluation of the Hawking energy on area-constrained critical surfaces under the dominant energy condition.
result The Hawking energy is nonnegative and rigid on area-constrained surfaces, including charged and cosmological constant variants.
Derives monotonic quantities for p-harmonic functions on manifolds.
problem Understanding p-harmonic functions on manifolds with nonnegative scalar curvature. method Derives local and global monotonic quantities associated with p-harmonic functions. result Establishes inequalities relating mass, capacity, and Willmore functional.
Study of Hawkes processes in limit order books for price volatility analysis.
problem Understanding price volatility in limit order books.
method Construct and analyze general compound Hawkes processes.
result Established Law of Large Numbers and Functional Central Limit Theorems for specific variations.
We provide estimates on the Bartnik mass of constant mean curvature (CMC) surfaces which are diffeomorphic to spheres and have positive mean curvature. We prove that the Bartnik mass is bounded from above by the Hawking mass and a new notion we call the asphericity mass. The asphericity mass is defined by applying Hami…
The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with nonzero mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch-Hawking mass. Th…
The study examines Hawkes processes and their long-term behavior.
problem Understanding the long-term behavior of Hawkes processes.
method Proving functional limit theorems under various conditions on the dispersion of child events.
result Functional limit theorems hold for Hawkes processes with different levels of child event dispersion.
The paper studies Hawkes processes under mean-field limits and criticality conditions.
problem Analyzing nearly unstable Hawkes processes in a mean-field regime.
method Extending the method by Jaisson and Rosenbaum, establishing scaling limits and propagation of chaos.
result Scaling limits of Hawkes processes are stochastic Volterra diffusions of affine type, with three distinct limiting regimes.
It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the …
The study proves local rigidity of minimal 2-spheres in electrovacuum spacetimes.
problem Proving local rigidity of minimal 2-spheres in electrovacuum spacetimes under certain conditions.
method Analyzing electrovacuum spacetimes and using constraints on charged Hawking mass and area minimization.
result Local rigidity of minimal 2-spheres in electrovacuum spacetimes, with isometric neighborhoods to specific spacetimes.
The paper develops Hawkes-based models for LOB and applies them to European, spread, and basket option pricing.
problem Developing accurate models for pricing options in the context of limit order books (LOB).
method Introduces multivariate Hawkes processes and their limit theorems, applies to European, spread, and basket options.
result Hawkes-based models provide more market forecast information than classical models.
Proves Penrose inequality for specific asymptotically flat manifolds.
problem Proving Penrose inequality for certain types of manifolds.
method Developed a new approximation scheme for a flow and established monotonicity of a free boundary Hawking mass.
result Proved the Riemannian Penrose inequality for specified manifolds.
New foliations found for critical surfaces of Hawking energy, resolving discrepancies.
problem Finding consistent critical surfaces for the Hawking energy in non-totally geodesic spacelike hypersurfaces.
method Constructing a unique local foliation of area constrained critical surfaces of the Hawking energy in the general case of non-totally geodesic spacelike hypersurfaces.
result Discrepancy found in the small sphere limit of the Hawking energy, explained and resolved.
On a compact Riemannian manifold with boundary having positive mean curvature, a fundamental result of Shi and Tam states that, if the manifold has nonnegative scalar curvature and if the boundary is isometric to a strictly convex hypersurface in the Euclidean space, then the total mean curvature of the boundary is no …
The paper proves a Penrose inequality involving quasi-local mass and outer trapped surfaces.
problem Detecting and quantifying the presence of black holes in spacetime.
method Comparison theorem and quasi-local mass analysis.
result Established a Penrose inequality linking quasi-local mass and outer trapped surfaces.
Sharp bounds for anisotropic p-capacity of Euclidean compact sets derived using flow methods.
problem Sharp bounds for anisotropic p-capacity of Euclidean compact sets.
method Inverse anisotropic mean curvature flow (IAMCF) and anisotropic Hawking mass.
result Upper bounds for anisotropic p-capacity derived using flow methods.
The paper proves nonexistence of NNSC cobordism for Bartnik data under certain conditions.
problem Proving nonexistence of NNSC cobordism for Bartnik data (Σ1n−1,γ1,H1) and (Σ2n−1,γ2,H2). method Analyzing metrics γ1 and γ2 on Sn−1 with fixed mean curvature H1 and large enough H2 to prove nonexistence of NNSC cobordism. result Proves nonexistence of NNSC cobordism for Bartnik data under specific conditions.