The paper finds at least four prime closed characteristics on star-shaped hypersurfaces in 8D space.
problem Finding prime closed characteristics on compact star-shaped hypersurfaces in 8D space.
method Proved existence of at least four prime closed characteristics for non-degenerate C3 compact star-shaped hypersurfaces in R8 without prime closed characteristics of Maslov-type index -1. result Existence of at least four prime closed characteristics on compact star-shaped hypersurfaces in R8. The paper proves inequalities for star-shaped and F-mean convex hypersurfaces in Rn+1.
problem Proving geometric inequalities for specific types of hypersurfaces.
method Using anisotropic p-momentum, perimeter, and volume, the paper derives inequalities for star-shaped and F-mean convex hypersurfaces. result The Wulff shape of F is the unique minimizer of the corresponding functionals among all star-shaped and F-mean convex sets. In 1998 Smoczyk [Smo98] showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in R3. We prove in this paper that this is true for the mean curvature flow of star-shaped hypersurfaces in Rn+1 in arbitrary …
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.
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.
Study eigenvalues for special curvature equations on star-shaped surfaces.
problem Eigenvalue problem for prescribed curvature equations on star-shaped, k-convex hypersurfaces.
method Established existence of a unique eigenvalue and hypersurface through uniform estimates in p for Lp-type equations.
result Existence of a unique eigenvalue and associated hypersurface under certain conditions.
Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.
problem Understanding the evolution of hypersurfaces with capillary boundaries.
method Locally constrained mean curvature flow for star-shaped hypersurfaces in the half-space.
result Established new Alexandrov-Fenchel inequalities for convex hypersurfaces with capillary boundary.
The Willmore inequality is extended to star-shaped and mean-convex hypersurfaces in hyperbolic space.
problem Proving a geometric inequality for specific hypersurfaces in hyperbolic space.
method Inverse mean curvature flow to prove the inequality.
result The Willmore inequality is generalized to star-shaped and mean-convex hypersurfaces in hyperbolic space.
Proves optimal isoperimetric inequality in de Sitter space.
problem Optimal isoperimetric inequality for specific hypersurfaces in de Sitter space.
method Analyzes spacelike, compact, star-shaped, and 2-convex hypersurfaces in de Sitter space.
result Proves an optimal isoperimetric inequality for the specified hypersurfaces.
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 study proves the existence of k-convex hypersurfaces for specific curvature equations.
problem Proving the existence of k-convex hypersurfaces for Hessian curvature equations. method Combining a priori estimates with the continuity method, and establishing a constant rank theorem.
result Existence and uniqueness of k-convex hypersurfaces for both nonhomogeneous and homogeneous Hessian curvature equations. 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…
Study introduces a new mean curvature flow for spherical boundaries.
problem Mean curvature flow for free boundary hypersurfaces in a ball.
method Guan-Li type volume preserving mean curvature flow for star-shaped free boundary hypersurfaces.
result The flow converges to a free boundary spherical cap for star-shaped initial data.
Smooth convergence to a sphere from cone-shaped surfaces.
problem Evolution of surfaces inside a cone using inverse mean curvature flow.
method Inverse mean curvature flow applied to star-shaped hypersurfaces meeting a convex cone perpendicularly.
result Smooth convergence to a sphere over time.
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.
In this paper, we use the inverse curvature flow to prove a sharp geometric inequality on star-shaped and two-convex hypersurface in hyperbolic space.
The study shows infinitely many Reeb orbits on star-shaped hypersurfaces with growth rate like prime numbers.
problem Growth rate of Reeb orbits on star-shaped hypersurfaces.
method Analyzing fiberwise star-shaped hypersurfaces in cotangent bundles with topological conditions.
result The number of Reeb orbits with period at most T grows at least like T/log(T).
We define a new version of modified mean curvature flow (MMCF) in hyperbolic space Hn+1, which interestingly turns out to be the natural negative L2-gradient flow of the energy functional defined by De Silva and Spruck in \cite{DS09}. We show the existence, uniqueness and convergence of the MMCF of com…
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…
In this paper we consider a star-shaped hypersurface flow by mean curvature. Without any assumption on the convexity, we give a new proof of gradient estimate for a short time. As an application, we also give a lower bound for the blowing up time.
Sharp bounds for Steklov eigenvalues on star-shaped domains and paraboloids.
problem Finding sharp lower bounds for Steklov eigenvalues on specific domains.
method Analyzing star-shaped bounded domains and paraboloids, using the largest geodesic ball as a reference.
result Sharp lower bounds for Steklov eigenvalues on star-shaped domains and paraboloids.
The paper studies curvature flows of star-shaped hypersurfaces and proves convergence to spheres.
problem Analyzing the convergence of a class of anisotropic curvature flows.
method Using new auxiliary functions, the paper studies a class of flows with specific speed and proves convergence under certain conditions.
result The k-convex solution to the flow converges smoothly to a sphere after normalization for specific values of k, α, and β. 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 study examines evolving star-shaped hypersurfaces in hyperbolic spaces, influenced by ambient geometry.
problem Evolution of star-shaped hypersurfaces in hyperbolic spaces.
method Nonhomogeneous expanding curvature flows in hyperbolic spaces.
result The asymptotic behavior of the flow depends on the ambient space's geometry, leading to different limiting metrics.
The study uses symplectic capacities to bound the systole on the sphere.
problem Bounding the systole on the sphere using symplectic capacities.
method Using symplectic capacities and properties of fiberwise balanced hypersurfaces.
result Upper bounds on the systole in terms of geometric data and β. 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.
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…
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.
Paper estimates curvature of convex hypersurfaces with prescribed curvature.
problem Estimating curvature of p-convex hypersurfaces with prescribed curvature. method Establishes curvature estimates for p-convex hypersurfaces in Rn+1 with p≥2n. result Proves existence of star-shaped hypersurface of prescribed curvature and interior C2 estimates. Study anisotropic flow for capillary hypersurfaces, proving new inequalities.
problem Anisotropic capillary hypersurfaces and their properties.
method Anisotropic volume-preserving mean curvature flow, new approach for strictly convex initial hypersurfaces.
result Established new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces.
The paper proves new inequalities in hyperbolic space using Euclidean methods.
problem Proving weighted isoperimetric inequalities in hyperbolic space.
method Using isoperimetric inequality with log-convex density in Euclidean space.
result Removed horo-convex assumption and proved new inequalities for star-shaped domains.
Study the isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.
problem Isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.
method Introduced a mean curvature type flow to study the isoperimetric problem.
result Established the isoperimetric inequality for star-shaped hypersurfaces in such manifolds.
The paper studies a curvature flow on hypersurfaces in R^(n+1).
problem Analyzing the long-term behavior of a specific type of curvature flow.
method Examining a flow defined by a non-homogeneous anisotropic speed function.
result The flow converges to a sphere for star-shaped and k-convex initial hypersurfaces.
We consider a compact, star-shaped, mean convex hypersurface Σ2⊂R3. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \cite{An1}). We also prove that in the case we have a surface of revolution which …
We show that the constant mean curvature hypersurfaces in the hyperbolic n-space spanning the boundary of a star shaped C^{1,1} domain in the asymptotic sphere give a foliation of the hyperbolic n-space. We also show that if C is a closed codimension-1 C^{2,a} submanifold in the asymptotic sphere bounding a unique cons…
Study proves a new formula for capillary hypersurfaces and shows a flow converging to a special shape.
problem Understanding the behavior of capillary hypersurfaces in hyperbolic space.
method Developed a volume-preserving flow starting from a star-shaped initial hypersurface and proved its long-time existence and convergence.
result The flow converges to a θ-totally umbilical cap, which is an energy minimizer for a given enclosed volume. In this paper, we consider the inverse hessian quotient curvature flow with star-shaped initial hypersurface in anti-de Sitter-Schwarzschild manifold. We prove that the solution exists for all time, and the second fundamental form converges to identity exponentially fast.
In this work, we prove an optimal Penrose inequality for asymptotically locally hyperbolic manifolds which can be realized as graphs over Kottler space. Such inequality relies heavily on an optimal weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Kottler space.
Proves inequalities for convex hypersurfaces in Euclidean space.
problem Geometric inequalities for hypersurfaces in Euclidean space.
method Proves two weighted geometric inequalities involving area, volume, and mean curvature.
result Examples of convex surfaces with ratios less than the round sphere.
New Minkowski inequality for capillary surfaces in half-space.
problem Establishing a new Minkowski inequality for capillary surfaces.
method Flow of inverse mean curvature type for capillary hypersurfaces in a half-space.
result Derive a new Minkowski-type inequality for star-shaped and mean convex capillary hypersurfaces.
We consider the inverse curvature flows x˙=F−pν of closed star-shaped hypersurfaces in Euclidean space in case 0<p=1 and prove that the flow exists for all time and converges to infinity, if 0<p<1, while in case p>1, the flow blows up in finite time, and where we assume the initial hypersurface to be…
We consider inverse curvature flows in $\Hh$ with star-shaped initial hypersurfaces and prove that the flows exist for all time, and that the leaves converge to infinity, become strongly convex exponentially fast and also more and more totally umbilic. After an appropriate rescaling the leaves converge in C∞ to…
Paper studies Hessian quotient equations in warped product manifolds.
problem Analyzing Hessian quotient equations in warped product manifolds.
method Using standard degree theory and a priori estimates.
result Existence of star-shaped compact hypersurface solutions.
The flow of symmetric spheres converges to a round sphere.
problem The behavior of symmetric hypersurfaces under inverse mean curvature flow.
method Localized parabolic maximum principle approach.
result The flow homothetically converges to a round sphere.
We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped (n−k)-convex bodies with prescribed k-th curvature measures (k>0) has been a longstanding problem. This is settled in this paper through the establishment of a crucial C2 a priori estimate for the c…
Introduces Star-Shaped deviation measures for risk analysis.
problem Risk measurement and analysis in finance.
method Characterizes Star-Shaped deviation measures through acceptance sets and convex deviation measures.
result Exposes the relationship between Star-Shaped risk measures and deviation measures.
Proves existence of static vacuum metrics with specific boundary data.
problem Existence of static vacuum metrics with prescribed boundary data.
method Proves existence and local uniqueness of static vacuum metrics close to the Euclidean metric.
result Existence of static vacuum metrics with prescribed Bartnik boundary data.
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.