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.
Inverse mean curvature flows studied in warped product manifolds with positive warping factor.
problem Analyzing inverse mean curvature flows in warped product manifolds.
method Investigating starshaped, mean convex hypersurfaces in warped product manifolds with positive warping factor.
result Existence and properties of inverse mean curvature flows in warped product manifolds with positive warping factor.
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.
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.
Study inverse mean curvature flow in quaternionic hyperbolic space, proving flow properties and convergence.
problem Evolution of star-shaped hypersurfaces in quaternionic hyperbolic space.
method Inverse mean curvature flow, star-shaped hypersurface, mean convex, convergence analysis.
result Flow is defined for any positive time, evolving hypersurface stays star-shaped and mean convex, induced metric converges to a conformal multiple of the standard sub-Riemannian metric on the sphere.
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 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.
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…
Investigates conditions for existence of inverse mean curvature flow in warped product manifolds.
problem Ensuring existence of inverse mean curvature flow in warped product manifolds.
method Analyzes sufficient curvature conditions for the Riemannian metric.
result Determines conditions on curvature for flow existence.
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. The paper studies self-expanding solutions to inverse curvature flows in Euclidean spaces.
problem Investigating self-expanding solutions to inverse curvature flows in Euclidean spaces.
method Using homogeneous symmetric functions of principal curvatures, the paper analyzes self-expanding solutions to a broad class of inverse curvature flows.
result Complete non-compact self-expanders to these flows with asymptotically cylindrical ends must be rotationally symmetric.
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…
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. The paper studies how shapes evolve in complex hyperbolic space.
problem Evolution of shapes in complex hyperbolic space.
method Inverse mean curvature flow applied to star-shaped, mean convex hypersurfaces.
result The flow is defined for any positive time, and the evolving shape remains star-shaped and mean convex.
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 [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…
The paper classifies shapes of translating solitons for a specific flow.
problem Understanding the shapes of translating solitons in a specific flow.
method Analyzing functions on a unit sphere and solving an ODE.
result Classification of the shapes of translating solitons.
We prove that the leaves of an inverse mean curvature flow provide a foliation of a future end of a cosmological spacetime N under the necessary and sufficent assumptions that N satisfies a future mean curvature barrier condition and a strong volume decay condition. Moreover, the flow parameter t can be used to d…
The paper classifies ruled surfaces in a specific space that move in a special way.
problem Understanding how surfaces move in a special space.
method Classifying ruled surfaces as translating solitons of inverse mean curvature flow.
result Existence of non-cylindrical ruled translating solitons.
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 …
Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.
problem Understanding the asymptotic behavior of anisotropic mean curvature flow.
method Established local gradient estimates for anisotropic p-harmonic functions and weak solutions of IAMCF. result Weak IAMCF is asymptotic to the expanding Wulff shape solution at infinity.
This paper extends IMCF theory to Heisenberg group, solving Penrose inequality.
problem Extending inverse mean curvature flow theory to Heisenberg group.
method Developed a sub-Riemannian theory for Heisenberg group, introduced a flow preserving H-perimeter.
result Established a Minkowski-type formula in Heisenberg group, proving Heintze-Karcher inequality.
Paper classifies ruled surfaces in Lorentz-Minkowski space for a specific flow.
problem Classifying ruled surfaces in Lorentz-Minkowski space.
method Examining homothetic self-similar solutions of the inverse mean curvature flow.
result Existence of two classes of non-cylindrical homothetic solitons.
We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in Rn. As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the volume enclosed by a hypersurface to a weighted total mean curvature of the hypers…
Unified view of monotonicity formulas for inverse mean curvature flow and p-capacitary potentials.
problem Understanding monotonicity formulas for various geometric flows and potentials.
method Refined analysis of p-capacitary potentials and their level sets. result Strong convergence of p-capacitary potentials to inverse mean curvature flow and curvature varifolds. 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 introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed usin…
New proof shows 3-manifolds with Ricci curvature bound are either flat or grow non-Euclidean.
problem Proving properties of 3-manifolds with Ricci curvature bounds.
method Inverse mean curvature flow approach.
result Proves that 3-manifolds are either flat or have non-Euclidean volume growth.
Modeling bone microarchitecture adaptation using geometric flows.
problem Bone microarchitecture adaptation modeling.
method Advection and mean curvature flow model with a sphere as a test case.
result Closed-form solution for sphere under advection and mean curvature flow.
We study solutions to the inverse mean curvature flow which evolve by homotheties of a given submanifold with arbitrary dimension and codimension. We first show that the closed ones are necessarily spherical minimal immersions and so we reveal the strong rigidity of the Clifford torus in this setting. Mainly we focus o…
In this paper, we show that the inverse anisotropic mean curvature flow in Rn+1, initiating from a star-shaped, strictly F-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially fast to a rescaled Wulff shape in the C∞ topology. As an application, we p…
We show that any complete, immersed self-expander to the inverse mean curvature flow, which has one end asymptotic to a cylinder, or has two ends asymptotic to two coaxial cylinders, must be rotationally symmetric.
Study inverse mean curvature flow on non-compact hypersurfaces, proving long-term existence and characterizing maximal time.
problem Evolution of non-compact convex hypersurfaces in Rn+1 by inverse mean curvature. method Establish long-term existence via pointwise mean curvature estimate and viscosity solutions for strict convexity.
result Characterization of maximal time of existence in terms of tangent cone at infinity.
Maps from spheres and disks to convex shapes via curvature flow.
problem Constructing contractions from spheres and disks to convex shapes.
method Inverse mean curvature flow to create normalized-area-preserving contractions.
result Proves E. Milman's conjecture and gives spectral comparison results.
We consider the smooth inverse mean curvature flow of strictly convex hypersurfaces with boundary embedded in Rn+1, which are perpendicular to the unit sphere from the inside. We prove that the flow hypersurfaces converge to the embedding of a flat disk in the norm of C1,β, β<1.
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.
Study inverse mean curvature flow on entire graphs, proving finite time existence for certain asymptotic cases.
problem Analyzing the evolution of entire graphs under inverse mean curvature flow.
method Global existence for starshaped graphs, critical case analysis for asymptotically conical graphs.
result Existence of a finite time \( T \) for certain asymptotically conical graphs, convergence to a flat plane as \( t o T \).
By studying the monotonicity of the first nonzero eigenvalues of Laplace and p-Laplace operators on a closed convex hypersurface Mn which evolves under inverse mean curvature flow in Rn+1, the isoperimetric lower bounds for both eigenvalues were founded.
The paper proves new Minkowski inequalities for flows in warped spaces.
problem Proving new Minkowski inequalities for flows in warped spaces.
method Locally constrained inverse curvature flows in Riemannian warped spaces.
result New Minkowski inequalities are derived for flows in warped spaces.