3D spherical caps are rigid under certain perturbations.
problem Rigidity of 3D spherical caps under specific perturbations.
method Gromov's μ-bubble technique
result 3D spherical caps are rigid under perturbations that maintain metric, scalar curvature, and mean curvature.
Study of free boundary minimal Möbius bands in spherical caps.
problem Characterizing minimal surfaces with free boundary in spherical caps.
method Analyzing spectral properties and geometric constraints.
result Proves that any free boundary minimal Möbius band in spherical caps must be intrinsically rotationally symmetric.
Proves rigidity for spherical cap eigenvalue problem.
problem Eigenvalue problem with mixed boundary conditions.
method Obata-type rigidity result for spherical cap.
result Proves rigidity for eigenvalue problem.
The study finds the optimal metrics for free boundary minimal surfaces in spherical caps.
problem Optimizing metrics for free boundary minimal surfaces in spherical caps.
method Introducing functionals based on eigenvalues of Steklov-type problems and proving maximizers are induced by immersions.
result Maximizing metrics are induced by free boundary minimal immersions in geodesic balls of a round sphere.
Study a flow in a ball that preserves volume and converges to spherical caps.
problem Preserving volume in a flow with a capillary boundary.
method Mean curvature flow with capillary boundary.
result The flow has longtime existence and converges to spherical caps.
New flow for capillary surfaces converges to spherical caps.
problem Optimizing capillary surfaces in space forms.
method Constrained mean curvature flow.
result Flow converges to spherical caps globally.
The study proves rigidity and non-rigidity of spherical caps in mean curvature.
problem Understanding mean curvature rigidity and non-rigidity on spherical caps.
method Used a Tangency Principle to prove rigidity and constructed counterexamples for non-rigidity.
result Contrast between rigidity and non-rigidity phenomena on spherical caps.
Study investigates minimal surfaces in spherical caps, extending previous findings.
problem Characterizing minimal surfaces with free boundaries and capillary conditions in spherical caps.
method Extending previous half-space intersection properties to warped products and capillary minimal surfaces in high codimension.
result Established a dual operation relating free boundary and capillary minimal surfaces.
Study finds surfaces in spherical caps that maximize modified energy.
problem Geometry of surfaces with free boundaries and capillary conditions.
method Monotonicity formulae and energy maximization analysis.
result Capillary minimal surfaces maximize a modified energy in their conformal orbit.
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.
Study spherical cap packing with probabilistic methods for detecting low-rank structures.
problem Detecting low-rank structures in high-dimensional Gaussian data.
method Probabilistic spherical cap packing approach for asymptotic bounds and extreme value distributions.
result Developed fast detection method for low-rank structures without spectrum information.
Study convex capillary hypersurfaces with prescribed curvature in a spherical cap.
problem Prescribed curvature problem for convex capillary hypersurfaces.
method Reformulated as Hessian quotient equation with Robin boundary condition.
result Existence of strictly convex capillary hypersurface with prescribed curvature.
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.
Study shows how curved surfaces evolve smoothly to spherical shapes.
problem Evolution of curved surfaces with capillary boundaries.
method Volume-preserving curvature flow with power mean curvature speed.
result Convex initial hypersurfaces evolve to spherical caps over time.
It is known that planar disks and small spherical caps are the only constant mean curvature graphs whose boundary is a round circle. Usually, the proof invokes the Maximum Principle for elliptic equations. This paper presents a new proof of this result motivated by an article due to Reilly. Our proof utilizes a flux fo…
Proves nonnegatively curved hypersurfaces on a sphere are convex disks.
problem Characterizing nonnegatively curved hypersurfaces with free boundary on a sphere.
method Analyzes hypersurfaces in Euclidean space with constant mth mean curvature. result Compact hypersurfaces are embedded convex disks.
Paper solves inequalities for capillary hypersurfaces in half-spaces.
problem Finding inequalities for convex capillary hypersurfaces in half-spaces.
method Introduced quermassintegrals and constructed a new locally constrained curvature flow to prove convergence to spherical caps.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
We give lower and upper bounds for the first eigenvalue of geodesic balls in spherically symmetric manifolds. These lower and upper bounds are C0-dependent on the metric coefficients. It gives better lower bounds for the first eigenvalue of spherical caps than those from Betz-Camera-Gzyl.
New degenerate free boundary minimal annuli found in spherical caps, challenging uniqueness.
problem Non-uniqueness in spherical caps beyond the hemisphere.
method Analyzing a family of embedded free boundary minimal annuli in geodesic balls.
result Degenerate annuli exist, contradicting the Naff-Zhu uniqueness hypothesis.
The paper derives Pizzetti formulae and inverts the Radon transform on spheres.
problem Inverting the Radon transform on spheres.
method Obtained Pizzetti-type formulae on sphere regions, used delta distributions, and derived inversion formulae.
result Derived Pizzetti formulae and inversion formulae for the Radon transform on spheres.
Paper solves capillary Orlicz-Minkowski problem with new inequalities.
problem Finding capillary convex bodies with prescribed Orlicz surface area measures.
method Continuity method and inequalities to solve the capillary even Orlicz-Minkowski problem.
result Volume-normalized smooth solutions and inequalities established.
The paper shows how to use hyperplanes and hyperballs interchangeably using inversive geometry.
problem Tackles the interchangeability of hyperplanes and hyperballs in discriminative boundaries.
method Applies inversive geometry to transform Euclidean data into spherical data and back, providing explicit formulae.
result Shows a duality between hyperspherical caps and hyperballs, providing explicit formulae to map between them.
The paper proves sphere theorems for charged bodies in linear potential theory.
problem Analyzing the capacitary potential of a charged body to deduce geometric inequalities.
method Analyzing the mean curvature and applying inequalities to domains with spherical symmetry.
result Domains with spherical symmetry are the only ones satisfying the given curvature condition.
Study stable capillary hypersurfaces with planar boundaries in half-spaces and domains.
problem Characterize stable capillary hypersurfaces with planar boundaries in bounded domains.
method Analyzes hypersurfaces in half-spaces and domains bounded by hyperplanes, proving conditions for stability and shape.
result Stable hypersurfaces in certain domains are spherical caps or pieces of spheres.
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)−sphere. Optimal inequality on sphere for convex bodies.
problem Bounding the spherical measure of a convex body's intersection with a plane orthogonal to its centroid.
method Proving an inequality on the sphere using convex geometry.
result The inequality is optimal with a constant of \((1-1/n)^{n-1}\).
Proves stability of convex disks close to round caps.
problem Stability of convex disks with positive curvature and strictly convex boundary.
method Compactness result for a Liouville-type PDE problem.
result Proves stability for a theorem of F. Hang and X. Wang.
The paper proves inequalities for convex capillary hypersurfaces in a half-space.
problem Proving inequalities for convex capillary hypersurfaces in a half-space.
method Locally constrained inverse curvature flow with spherical cap convergence.
result Proves a complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
Study of metrics with prescribed curvature and geodesic curvature on a disc.
problem Existence and behavior of conformal metrics with prescribed curvature and boundary geodesic curvature.
method Variational characterization and gradient flow approach.
result Existence of solutions or blow-up to a spherical cap, leading to existence results via shadow flow.
The abstract constructs a set of bad 3-orbifolds and shows how any bad 3-orbifold can be transformed into a good one.
problem Characterizing and transforming bad 3-orbifolds into good ones.
method Explicit construction of bad 3-orbifolds and a method of cutting-and-capping to transform them.
result Any bad 3-orbifold can be transformed into a good 3-orbifold through a finite number of operations.
We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Be…
We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the origin, then the hypersurface is unstable. An immediate result is that all known ex…
3-manifolds with hyperbolic handlebody complements are studied.
problem Understanding the geometry of 3-manifolds with hyperbolic handlebody complements.
method Capping off spherical and torus boundaries, constructing handlebodies, and applying octahedral decomposition.
result Bounds on volume for some handlebody complements are derived.
Paper proves rigidity for compact manifolds with boundary.
problem Rigidity of compact connected locally conformally flat manifolds.
method Analyzes fully nonlinear elliptic conformally invariant equations.
result Compact connected locally conformally flat manifolds are isometric to closed geodesic balls.
Fixed points of mean section operators found in convex bodies.
problem Characterizing fixed points of mean section operators in convex bodies.
method Characterization of rotation equivariant operators using spherical Laplacian mass distribution, and application of Minkowski valuations.
result Euclidean balls are the only fixed points of mean section operators in a C2 neighborhood of the unit ball. 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.
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.
New proof that certain stable surfaces can't have genus 1.
problem Characterizing stable constant mean curvature surfaces with free boundary.
method Modified Hersch type balancing argument.
result Proves that surfaces cannot have genus 1.
The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n > 2. In this paper we describe a necessary condition that th…
We show that a capillary surface in a solid cone, that is, a surface that has constant mean curvature and the boundary of surface meets the boundary of the cone with a constant angle, is radially graphical if the mean curvature is non-positive with respect to the Gauss map pointing toward the domain bounded by the surf…
The paper solves a specific Minkowski problem for capillary hypersurfaces.
problem Finding capillary convex bodies with prescribed dual curvature measures.
method Reduction to a Monge-Ampère type equation with Robin boundary condition.
result Existence and uniqueness of a smooth solution for θ∈(0,2π). The paper proves new inequalities for convex hypersurfaces using centro-affine geometry.
problem Proving inequalities for convex hypersurfaces.
method Introducing a flat logarithmic centro-affine geometry and using Bochner formulas.
result Established new Poincaré and Brunn-Minkowski inequalities.
Study on convex capillary hypersurfaces with Lp curvature in half-space.
problem Prescribed Lp curvature for convex capillary hypersurfaces.
method Reduction to Hessian quotient equation with Robin boundary condition.
result Existence and uniqueness of smooth admissible solutions.
The paper uses LSMC to price capped American options with time-dependent caps.
problem Pricing American options with time-capped features.
method Least Squares Monte Carlo (LSMC) method.
result The LSMC method converges to the true price as discretization step and number of trajectories approach limits.
The paper examines ancient solutions of mean curvature flow in space forms with curvature pinching conditions.
problem Investigating rigidity of ancient solutions of mean curvature flow in space forms.
method Sharp asymptotic pointwise curvature pinching conditions and asymptotic integral curvature pinching conditions.
result Ancient solutions in a sphere are either a shrinking spherical cap or a totally geodesic sphere, and in a hyperbolic space, they are a family of shrinking spheres.
The study pinches conditions for constant mean curvature surfaces in convex 3-manifolds.
problem Understanding the topology and geometry of constant mean curvature surfaces with free boundaries in convex 3-manifolds.
method Provided pinching conditions on the traceless second fundamental form to guarantee surface topology.
result The surface is either a disk, annulus, spherical cap, or Delaunay surface under certain conditions.
Method finds all cross caps formally isometric to a given one.
problem Identifying cross caps that are formally isometric to a given one.
method Finding cross caps with matching Taylor expansions of first fundamental forms.
result A countable family of intrinsic invariants recognizes formal isometry classes completely.
It is classically known that generic smooth maps of R^2 into R^3 admit only cross cap singularities. This suggests that the class of cross caps might be an important object in differential geometry. We show that the standard cross cap (u,uv,v^2) has non-trivial isometric deformations with infinite dimensional freedom. …