The paper proves the existence of a hyperbolic inverse mean curvature flow under specific conditions.
problem Proving the existence of a hyperbolic inverse mean curvature flow.
method Short-time existence proof under mean convex and star-shaped initial conditions.
result Short-time existence of hyperbolic inverse mean curvature flow under specified conditions.
Inverse mean curvature flow converges to a disk in hyperbolic space.
problem Understanding flow behavior in hyperbolic geometry.
method Inverse mean curvature flow with free boundary on geodesic spheres.
result Flow converges to a totally geodesic disk.
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.
This note revisits the inverse mean curvature flow in the 3-dimensional hyperbolic space. In particular, we show that the limiting shape is not necessarily round after scaling, thus resolving an inconsistency in the literature.
We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose--type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow …
The paper proves constant mean curvature surfaces in specific manifold types.
problem Existence of surfaces with constant mean curvature in asymptotically flat and hyperbolic manifolds.
method Combines min-max theory with inverse mean curvature flow.
result Existence of compact surfaces with constant mean curvature in asymptotically flat and hyperbolic manifolds.
In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the quaternionic hyperbolic space. We p…
We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to…
Proves a Minkowski inequality for star-shaped hypersurfaces in warped cylinders.
problem Proving a Minkowski inequality for specific types of hypersurfaces.
method Using weakly mean convex and star-shaped hypersurfaces in warped cylinders, and applying the inverse mean curvature flow.
result Sharp inequality holds for outward minimizing hypersurfaces in Schwarzschild and hyperbolic spaces.
The article uses harmonic mean curvature flow to prove new geometric inequalities for convex hypersurfaces.
problem Proving new geometric inequalities for convex hypersurfaces in hyperbolic space.
method Harmonic mean curvature flow, Alexandrov-Fenchel inequalities, inverse mean curvature flow, Heintze-Karcher type inequality.
result New geometric inequalities for convex hypersurfaces in hyperbolic space.
Sharp upper bounds derived for capacities in hyperbolic and Euclidean spaces.
problem Finding upper limits for the capacity of compact sets in hyperbolic and Euclidean spaces.
method Inverse mean curvature flow, unit-speed normal flow, weak inverse mean curvature flow, inverse anisotropic mean curvature flow.
result Various sharp upper bounds for the p-capacity of compact sets in hyperbolic and Euclidean spaces are derived. In [8] Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Further…
New inequalities derived for hyperbolic space via specific flows.
problem Sharp inequalities for mean and k-th mean curvatures in hyperbolic space.
method Locally constrained inverse curvature flow by Brendle, Guan, and Li.
result Established and verified new sharp inequalities for hyperbolic space.
The paper studies a flow of spacelike surfaces in Lorentz-Minkowski space, proving convergence to a hyperbolic plane.
problem Evolution of spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Anisotropic inverse mean curvature flow with Neumann boundary condition.
result The flow converges to a hyperbolic plane as time tends to infinity.
The paper studies how certain spacelike surfaces evolve over time in a specific space.
problem Evolution of spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Inverse mean curvature flow with vanishing Neumann boundary condition.
result The evolving surfaces converge to a hyperbolic plane as time goes to infinity.
We investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over horospheres in Hn+1 and show long time existence of the flow. Along the way many important local estimates as well as global estimates are obtained. In addition,…
Paper proves Penrose inequality for graphs over specific spacetime manifolds.
problem Proving Penrose inequality for graphs over Reissner-Nordström-anti-deSitter manifold.
method Inverse mean curvature flow to establish optimal Minkowski type inequality and Penrose type inequality.
result Established Penrose type inequality for graphs over Reissner-Nordström-anti-deSitter manifold.
The paper studies inverse mean curvature flow on hypersurfaces in space forms.
problem Analyzing the inverse mean curvature flow on hypersurfaces in space forms.
method Investigates the existence and properties of the flow for isoparametric hypersurfaces.
result Characterizes the flow and solutions explicitly under certain conditions.
In this article, we prove a geometric inequality for star-shaped and mean-convex hypersurfaces in hyperbolic space by inverse mean curvature flow. This inequality can be considered as a generalization of Willmore inequality for closed surface in hyperbolic 3-space.
Study stability of mass theorems for hyperbolic manifolds foliated by IMCF.
problem Stability of Positive Mass Theorem and Riemannian Penrose Inequality in asymptotically hyperbolic manifolds.
method Sequence of regions foliated by IMCF, convergence to hyperbolic or AdS-Schwarzschild metrics.
result Convergence of regions to specific metrics under given conditions.
New flow expands hypersurfaces in hyperbolic space, showing round limiting shape for certain powers.
problem Understanding the limiting shape of hypersurfaces expanding in hyperbolic space.
method Introduced shifted inverse curvature flow with positive power p for a smooth curvature function. result For 0<p≤1, limiting shape is always round as maximal existence time is approached. Proves smoothness and star-shapedness of weak IMCF solutions in hyperbolic space.
problem Analyzing weak inverse mean curvature flow in hyperbolic space.
method Inspired by Alexandrov reflection method, uses Li-Wei result.
result Proves expanding spheres as the only proper weak IMCF on hyperbolic space.
The article uses inverse mean curvature flow to prove inequalities for hypersurfaces in hyperbolic space.
problem Establishing optimal Sobolev-type inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space.
method Inverse mean curvature flow to prove inequalities.
result Proves hyperbolic Alexandrov-Fenchel inequalities and optimal quermassintegral inequalities.
The paper proves Michael-Simon inequalities in hyperbolic space using novel curvature flows.
problem Proving the sharp Michael-Simon inequality for mean curvature in hyperbolic space.
method Developed new locally constrained curvature flows for proving the inequality.
result Sharp Michael-Simon inequalities for mean and k-th mean curvatures in starshaped hypersurfaces in hyperbolic space.
Proves existence of proper solutions for inverse mean curvature flow.
problem Existence of proper solutions for inverse mean curvature flow.
method Proves existence theorem assuming non-degeneracy conditions on isoperimetric profile.
result No curvature assumption in existence theorem.
We consider inverse curvature flows in hyperbolic space with starshaped initial hypersurface, driven by positive powers of a homogeneous curvature function. The solutions exist for all time and, after rescaling, converge to a sphere.
We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form F−p, where p>1 and F is a positive, strictly monotone and 1-homogeneous curvature function. In particular this class includes the mean curvature F=H. We prove that a certain initial…
Proves higher regularity for anisotropic inverse mean curvature flow.
problem Higher regularity of solutions to anisotropic inverse mean curvature flow.
method Proves Harnack estimate and constructs smooth solutions from C1 initial sets. result Smooth solutions become smooth outside a compact set.
Study constructs solutions for evolving hypersurfaces using inverse spacetime mean curvature.
problem Evolution of hypersurfaces in spacetime.
method Weak solutions for hypersurfaces evolving along inverse spacetime mean curvature in asymptotically flat maximal initial data sets.
result Weak solution detects both future- and past-trapped apparent horizons.
The paper lists all self-similar solutions for a flow in 2D space.
problem Finding solutions to the inverse mean curvature flow in 2D.
method Obtained a complete list of self-similar solutions.
result Completely enumerated all self-similar solutions for the flow.
Proves uniqueness of geometric flow in various Riemannian manifolds.
problem Proving uniqueness of geometric flow in general Riemannian manifolds.
method Two backward uniqueness theorems for extrinsic geometric flow.
result Backward uniqueness of extrinsic geometric flow in general ambient manifolds.
The paper studies how singularities evolve in inverse mean curvature flow.
problem Understanding the evolution of singularities in inverse mean curvature flow.
method Defining a weak solution, analyzing blow-up tangent cones, and proving the evolution of singularities.
result Each singularity is removed when the evolving cone becomes flat, leading to the exact waiting time for a weak solution to be smooth.
In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface Σ is strictly mean convex and star-shaped, then the flow hypersurface Σt converges to a large coordinate sphere as t→∞ exponentially. We also describe an a…
Inverse Mean Curvature Flow bounds star-shaped surfaces' lifetime.
problem Bounding lifetime of star-shaped surfaces under IMCF.
method Derives upper bound on waiting time for variational weak solutions.
result Star-shaped surfaces develop singularities or self-intersections within a prescribed time.
The paper proves existence and growth estimates for inverse mean curvature flow and related p-Laplacian Green kernel decay.
problem Existence and growth estimates for inverse mean curvature flow.
method Proving new decay estimates for the Green kernel of the p-Laplacian. result Existence and optimal growth estimates for the weak inverse mean curvature flow.
The flow of a torus by inverse mean curvature keeps total curvature bounded until singularity.
problem Understanding the behavior of a torus under inverse mean curvature flow until singularity.
method Analyzing the evolution of a rotationally symmetric embedded torus in R3 by inverse mean curvature flow. result The total curvature remains bounded until the singular time Tmax. 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.
We consider the inverse mean curvature flow in Robertson-Walker spacetimes that satisfy the Einstein equations and have a big crunch singularity and prove that under natural conditions the rescaled inverse mean curvature flow provides a smooth transition from big crunch to big bang. We also construct an example showing…
Ancient solutions found for mean curvature flows in hyperbolic spaces.
problem Mean curvature flows of isoparametric submanifolds in hyperbolic spaces.
method Study mean curvature flows of isoparametric submanifolds in hyperbolic spaces.
result Ancient solutions always exist for these flows.
The round sphere is the only solution to inverse curvature flows under certain conditions.
problem Proving uniqueness of self-conformal solutions to inverse curvature flows.
method Analyzing flows by diffeomorphisms generated by conformal Killing fields.
result The round sphere is the only closed solution to the inverse mean curvature flow and related flows under natural conditions.
New boundary condition for weak inverse mean curvature flow in bounded domains.
problem Addressing the well-posedness of inverse mean curvature flow in bounded domains with an outer obstacle.
method Developed a new boundary condition, combined techniques including elliptic regularization, blow-up analysis, and parabolic estimates.
result Existence and uniqueness theorem for weak solutions in smooth bounded domains, with C1,α regularity of level sets up to the obstacle. Develops Aleksandrov reflection for hyperbolic flows, proving convergence to umbilic surfaces.
problem Analyzing geometric flows in hyperbolic spaces.
method Aleksandrov reflection framework applied to level-set formulation, with graphical and Lipschitz estimates.
result Solutions converge exponentially fast to an umbilic hypersurface at infinity.
The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist …
Study on stability of Positive Mass Theorem using Inverse Mean Curvature Flow.
problem Stability of Positive Mass Theorem in foliated regions with positive scalar curvature.
method Analyzes sequences of regions foliated by solutions to Inverse Mean Curvature Flow, focusing on convergence to flat annuli under specific conditions.
result Convergence of foliated regions to flat annuli under certain conditions, leading to stability of Positive Mass Theorem.
In this work we extend the ODE Maximum principle of Hamilton to non-compact hypersurfaces using the Omari-Yau maximum principle at infinity. As an application of this result, we investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over …
We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor φ(r). If φ′(r)>0 and φ′′(r)≥0, we show that these flows exist for all times, remain starshaped and mean convex. Plus the positivity of φ′′(r) and …
Study inverse curvature flows with local constraints in warped manifolds.
problem Understanding geometric flows with local constraints in complex manifolds.
method Inverse curvature flows in warped product manifolds with local constraints.
result Longtime existence and smooth convergence to coordinate slices.
In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-…