Proves convexity of certain hypersurfaces with negative λ.
problem Understanding convexity of hypersurfaces with specific λ values.
method Analyzes mean convex hypersurfaces and proves convexity for λ ≤ 0.
result Closed n n n -dimensional mean convex λ λ λ -hypersurfaces are convex if λ ≤ 0 λ \le 0 λ ≤ 0 . Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.
problem Finding sharp bounds on total mean curvature of convex hypersurfaces.
method Sharp lower bounds for mean width and Birkhoff invariant, characterizing spheres.
result Generalization of Álvarez Paiva's result to convex hypersurfaces.
New compact mean convex hypersurfaces found for positive λ.
problem Finding compact embedded hypersurfaces for positive λ.
method Constructing compact mean convex hypersurfaces diffeomorphic to spheres.
result No compact convex embedded λ-hypersurfaces except a round sphere for λ > 0.
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.
The paper proves inequalities for star-shaped and F F F -mean convex hypersurfaces in R n + 1 \mathbb{R}^{n+1} R n + 1 .
problem Proving geometric inequalities for specific types of hypersurfaces.
method Using anisotropic p p p -momentum, perimeter, and volume, the paper derives inequalities for star-shaped and F F F -mean convex hypersurfaces. result The Wulff shape of F F F is the unique minimizer of the corresponding functionals among all star-shaped and F F F -mean convex sets. The paper proves a Minkowski inequality for star-shaped, mean convex hypersurfaces.
problem Proving a Minkowski inequality for specific hypersurfaces.
method Inverse anisotropic mean curvature flow, star-shaped, strictly F F F -mean convex hypersurfaces. result The flow converges exponentially fast to a rescaled Wulff shape.
The article proves finiteness results for 2D convex hypersurfaces using surgery on mean curvature flow.
problem Proving finiteness for 2D convex hypersurfaces in R n + 1 \mathbb{R}^{n+1} R n + 1 . method Using mean curvature flow with surgery for 2 convex hypersurfaces.
result Proves extrinsic finiteness results in the spirit of Cheeger's compactness theorem.
Convex hypersurfaces evolve to spheres under a specific flow.
problem Volume preserving nonhomogeneous mean curvature flow of convex hypersurfaces.
method Monotonicity of isoperimetric ratio, inner and outer radius control, maximum principle arguments.
result Closed convex hypersurfaces converge to round spheres.
Constructs a mean curvature flow with surgery for compact mean convex hypersurfaces.
problem Mean curvature flow with surgery for compact mean convex hypersurfaces.
method Topological surgeries performed by the flow itself through nondegenerate cylindrical singularities, adjusted at smooth times.
result Extends previous results for 2-convex flows and constructs a flow for compact mean convex hypersurfaces.
Mean curvature flow preserves convexity for star-shaped hypersurfaces.
problem Preserving convexity in mean curvature flow for star-shaped hypersurfaces.
method Analyzing blowup limits and proving genericity for star-shaped hypersurfaces in arbitrary dimensions.
result The mean curvature flow of star-shaped hypersurfaces preserves convexity.
The paper studies a flow for convex capillary hypersurfaces in a ball, proving smooth convergence to a spherical cap.
problem Analyzing the behavior of convex capillary hypersurfaces under mean curvature flow.
method Introduced mean curvature flow for hypersurfaces in the unit Euclidean ball with capillary boundary. Proved smooth convergence to a spherical cap for strictly convex initial hypersurfaces.
result The flow preserves strict convexity and converges smoothly to a spherical cap for all positive time.
Lower bounds for surface area and volume of convex hypersurfaces.
problem Establishing bounds for surface area and volume of convex hypersurfaces.
method Using displacement under continuous maps to establish lower bounds.
result Proves a lower bound for the volume of a Riemannian n-sphere in all dimensions.
The paper studies how convex hypersurfaces evolve under a specific speed function of the mean curvature.
problem Evolution of convex hypersurfaces under a non-homogeneous speed function.
method Evolution of a closed convex hypersurface in R n + 1 {\mathbb{R}}^{n+1} R n + 1 with a speed function depending only on the mean curvature. result The flow exists on a finite maximal interval, convexity is preserved, and the hypersurfaces shrink to a point.
Proves inequality for convex hypersurfaces in spheres.
problem Proving inequalities for convex hypersurfaces in spheres.
method Inverse mean curvature flow to find a monotone quantity.
result Proves Alexandrov-Fenchel-type inequality.
Geometric inequality for convex hypersurfaces in a sphere.
problem Proving a geometric inequality for convex hypersurfaces with boundary on a sphere.
method Inverse mean curvature flow with a free boundary perpendicular to the sphere.
result Proved a geometric inequality involving the Willmore energy for convex hypersurfaces of dimension n ≥ 3 n\geq 3 n ≥ 3 with boundary on the sphere. Study ancient solutions to free boundary mean curvature flow in convex manifolds.
problem Understanding ancient solutions to free boundary mean curvature flow in convex manifolds.
method Establish rigidity results and construct foliations to describe ancient solutions.
result Ancient solutions to free boundary mean curvature flow are rigid and exhaust all possibilities under certain conditions.
Study shows constant curvature convex hypersurfaces on hyperboloids are parts of hyperboloids.
problem Characterizing convex hypersurfaces with constant curvature on hyperboloids.
method Analyzing hypersurfaces with constant higher order mean curvature and constant boundary angle.
result Hypersurfaces with constant curvature on hyperboloids are parts of hyperboloids.
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.
Eigenvalues of convex hypersurfaces are bounded by flow.
problem Eigenvalue bounds for convex hypersurfaces.
method Inverse mean curvature flow on convex hypersurfaces.
result Monotonicity of eigenvalues under flow leads to isoperimetric bounds.
The paper extends Liebmann's Theorem to convex hypersurfaces with boundary.
problem Proving properties of convex hypersurfaces with boundary in Euclidean space.
method Analyzing locally convex, embedded, compact, connected CMC hypersurfaces bounded by a closed strictly convex submanifold.
result Spherical caps are the only such hypersurfaces with non-zero constant mean curvature bounded by a ( n − 1 ) − (n-1)- ( n − 1 ) − sphere. The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.
problem Conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.
method Proving Bernstein type theorems for entire convex graphical hypersurfaces with zero Gaussian curvature in Euclidean and Minkowski contexts.
result Zero Gaussian curvature convex hypersurfaces must be hyperplanes if the mean curvature goes to zero at infinity.
The paper studies a flow of convex hypersurfaces with a specific speed.
problem Preserving volume while deforming hypersurfaces in Euclidean space.
method Flow of closed convex hypersurfaces with speed based on k k k -th mean curvature and volume constraints. result The flow converges to a round sphere for strictly convex initial hypersurfaces without curvature pinching.
The paper studies how certain surfaces evolve over time.
problem Evolution of specific types of surfaces in high dimensions.
method Approximation by simpler problems to prove long-term existence.
result Long-time existence of the H α H^α H α -flow for specified surfaces. 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 R n + 1 \mathbb{R}^{n+1} R n + 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.
Proves new inequality for hyperbolic space hypersurfaces.
problem Finding inequalities for hypersurfaces in hyperbolic space.
method Proves a Heintze-Karcher type inequality for shifted mean convex hypersurfaces.
result Proves Alexandrov type theorem and uniqueness result for hypersurfaces.
Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
A possible evolution of a compact hypersurface in R^n by mean curvature past singularities is defined via the level set flow. In the case that the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. We obtain as a conse…
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.
We obtain estimates on both size and dimensions of the singular set at the first blow-up time of the mean curvature flow of hypersurfaces whose initial data is σ k σ_k σ k -convex.
Rotationally symmetric solutions for mean curvature flow.
problem Understanding singularities in mean curvature flow.
method Analyzing cylindrical and gradient estimates for translators.
result Rotationally symmetric solutions for a class of flows.
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.
A submanifold M m M^m M m of a Euclidean space R m + p R^{m+p} R m + p is said to have harmonic mean curvature vector field if Δ H ⃗ = 0 Δ\vec{H}=0 Δ H = 0 , where H ⃗ \vec{H} H is the mean curvature vector field of M ↪ R m + p M\hookrightarrow R^{m+p} M ↪ R m + p and Δ Δ Δ is the rough Laplacian on M M M . There is a conjecture named after Bangyen Chen which states that submanifolds o…
Paper solves inequalities for convex hypersurfaces with free boundary in a ball.
problem Finding inequalities for convex hypersurfaces with free boundary in a ball.
method Introduced quermassintegrals and used a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.
result Obtained new Alexandrov-Fenchel inequalities for convex free boundary hypersurfaces.
We show that closed hypersurfaces in Euclidean space with nonnegative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the k-th mean curvature, for k greater than 2, as we construct the counter-examples for all k greater than 2. Our proof relie…
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.
Study on curvature bounds for specific hypersurfaces in Anti-de Sitter space.
problem Bounding principal curvatures of constant mean curvature hypersurfaces.
method Generalized convex hull concept and quantitative estimates based on width.
result Explicit bounds on sectional curvature and quasiconformal dilatation.
The paper derives inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.
problem Geometric inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.
method Comparison formula via Reilly's identities; geometric inequalities derived.
result Sharp lower bound for total first mean curvature in dimension 3.
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.
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.
Paper studies how certain hypersurfaces collapse in Hilbert space.
problem Collapse of hypersurfaces in Hilbert space.
method Investigates mean curvature flow with specific conditions.
result Invariant hypersurfaces collapse to group orbits under flow.
Flow doesn't get wider near singularities if they're convex.
problem Preventing the fattening of surfaces during flow.
method Analyzing mean curvature flow with mean convex singularities.
result The level set flow of a mean convex initial surface doesn't get wider near singularities.
For any n n n -dimensional smooth manifold Σ Σ Σ , we show that all the singularities of the mean curvature flow with any initial mean convex hypersurface in Σ Σ Σ are cylindrical (of convex type) if the flow converges to a smooth hypersurface M ∞ M_{\infty} M ∞ (maybe empty) at infinity. Previously this was shown (i) for n ≤ 7 n\leq 7 n ≤ 7 ,…
In this paper we prove that the H^k (k is odd and larger than 2) mean curvature flow of a closed convex hypersurface can be extended over the maximal time provided that the total L^p integral of the mean curvature is finite for some p
Total curvatures of certain hypersurfaces are continuous.
problem Continuity of curvatures in geometric settings.
method Hausdorff distance for hypersurfaces and convex bodies in Riemannian manifolds and Cartan-Hadamard spaces.
result Total generalized mean curvatures are continuous.
New control on diameter and curvature for evolving surfaces.
problem Controlling the diameter and curvature of evolving surfaces under mean curvature flow.
method Detailed analysis of cylindrical regions under mean curvature flow.
result Intrinsic diameter stays uniformly controlled as surfaces approach first singular time.
In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface Σ Σ Σ is strictly mean convex and star-shaped, then the flow hypersurface Σ t Σ_t Σ t converges to a large coordinate sphere as t → ∞ t\rightarrow \infty t → ∞ exponentially. We also describe an a…
Ancient convex solutions to flow equations are limited to simple shapes.
problem Characterizing ancient convex solutions to flow equations.
method Analyzing mean curvature flow and curvature functions of convex hypersurfaces.
result Ancient convex solutions to flow equations are limited to spherical, cylindrical, or planar shapes.
Having in mind the well known model of Euclidean convex hypersurfaces [4], [5], and the ideas in [1] many authors defined and investigate convex hypersurfaces of a Riemannian manifold. As it was proved by the first author in [7], there follows the interdependence between convexity and Gauss curvature of the hypersurfac…