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.
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.
Study inverse curvature flows for capillary hypersurfaces in a unit ball.
problem Understanding the behavior of capillary hypersurfaces under inverse curvature flows.
method Investigate inverse curvature flows for strictly convex, capillary hypersurfaces in the unit Euclidean ball.
result Establish existence and convergence results for inverse curvature flows.
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.
New pinching estimates control curvature ratios in inverse curvature flows.
problem Controlling curvature ratios in inverse curvature flows.
method Proving pinching estimates for strictly convex hypersurfaces in space forms.
result Smooth convergence of the inverse curvature flow is proven.
The paper studies a flow of Legendre curves, generalizing the inverse curvature flow of regular curves.
problem Analyzing the inverse curvature flow of Legendre curves.
method Investigates the unique existence, monotonicity, and asymptotic behavior of the flow.
result The flow asymptotically converges to a self-similar solution, categorized by initial curve.
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.
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.
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.
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 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.
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. 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-…
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.
Paper proves stability of quermassintegral inequalities using inverse curvature flow.
problem Stability of quermassintegral inequalities for nearly spherical sets.
method Inverse curvature flow with special rescaling to study quermassintegral inequalities.
result Decreasing rate of k-th quermassintegral is faster than Fraenkel asymmetry for nearly spherical sets.
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.
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. Study on curvature flow in Minkowski space for cocompact hypersurfaces.
problem Investigating curvature flow in Minkowski space for cocompact hypersurfaces.
method Investigation of the cocompact inverse \(σ_k\) curvature flow in Minkowski space.
result Longtime existence and convergence of the curvature flow established.
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.
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.
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 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…
The paper studies area-preserving and length-preserving inverse curvature flow for planar curves with singularities.
problem Investigating the evolution of planar curves with singularities under area-preserving and length-preserving inverse curvature flow.
method Area-preserving and length-preserving inverse curvature flow for ℓ-convex Legendre curves. result The flow results in a circle for ℓ-convex Legendre curves, providing geometric inequalities. 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…
Study on flows in warped spaces without symmetry assumptions.
problem Analyzing curvature flows in non-symmetric warped spaces.
method Long-time existence and umbilicity estimates for flows with inverse speeds.
result Established long-time existence and umbilicity for flows in warped products.
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. 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.
Curvature flow and inverse curvature flow solutions on 2D light cone identified.
problem Identifying self-similar solutions to curvature flow and inverse curvature flow on 2D light cone.
method Proved correspondence between CF and ICF solutions, analyzed ellipses and hyperboles, and characterized self-similar solutions.
result Ellipses and hyperboles are the only curves evolving under homotheties on the 2D light cone.
We consider inverse curvature flows in the (n+1)-dimensional Euclidean space, n≥2, expanding by arbitrary negative powers of a 1-homogeneous, monotone curvature function F with some concavity properties. We obtain asymptotical roundness, meaning that circumradius minus inradius of the flow hypersurfaces decay…
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 flow on de Sitter space for convex hypersurfaces.
problem Behavior of locally constrained inverse curvature flow in de Sitter space.
method Analyze flow on de Sitter space with specific initial conditions and inequalities.
result Derive Alexandrov-Fenchel type inequalities.
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.
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 give a family of monotone quantities along smooth solutions to the inverse curvature flows in Euclidean spaces. We also derive a related geometric inequality for closed hypersurfaces with positive k-th mean curvature.
Inverse curvature flows shape star-shaped hypersurfaces into spheres.
problem Evolution of star-shaped hypersurfaces inside a convex cone.
method Inverse curvature flows, convexity of the cone, gradient and Hölder estimates.
result Hypersurfaces converge to a round sphere as time goes to 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.
We consider the inverse curvature flows in the anti-de Sitter-Schwarzschild manifold with star-shaped initial hypersurface, driven by the 1-homogeneous curvature function. We show that the solutions exist for all time and the principle curvatures of the hypersurface converges to 1 exponentially fast.
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 …
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.
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.
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. 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…
The paper deforms circle packings on surfaces with constant curvature.
problem Deforming circle packings on surfaces to constant curvature.
method Using Ge-Xu's α-flow to deform initial inversive distance circle packings.
result The inversive distance circle packing with constant α-curvature is unique under certain conditions.
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…
We show that strictly convex surfaces expanding by the inverse Gauss curvature flow converge to infinity in finite time. After appropriate rescaling, they converge to spheres. We describe the algorithm to find our main test function.