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.
Extends characterization of PD3-pairs with aspherical boundaries to those with spherical boundaries.
problem Characterizing fundamental triples of PD3-pairs with boundary components of different types. method Extends Turaev and Bleile's work by relaxing the π1-injectivity hypothesis and considering pairs with spherical boundary components. result Characterization of fundamental triples for PD3-pairs with spherical boundary components and c.d.π1(P)≤2. 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.
The study constructs free boundary CMC annuli in spherical and hyperbolic balls.
problem Finding free boundary CMC annuli in spherical and hyperbolic balls.
method Constructing free boundary CMC annuli with constant mean curvature H in geodesic balls of S^3 and H^3.
result Embedded free boundary CMC annuli exist for certain mean curvatures in both spaces.
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.
New minimal surfaces found with spherical curvature lines.
problem Finding minimal surfaces with specific curvature lines.
method Constructing surfaces parametrized by rhombic lattices.
result Found new examples of minimal annuli with free boundaries.
We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot comp…
Study on connectivity of Morse boundaries of Coxeter groups.
problem Connectivity of Morse boundaries of Coxeter groups.
method Defined conditions on defining graphs (wide-avoidant, wide-spherical-avoidant) and characterized Morse boundaries based on these conditions.
result Characterization of Morse boundary connectivity for different classes of Coxeter groups.
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.
Rigidity theorem for spherical sectors in Riemannian manifolds.
problem Rigidity of spherical sectors in Riemannian manifolds under overdetermined conditions.
method Analyzing solutions to the inhomogeneous Helmholtz equation with constant Dirichlet and Neumann boundary conditions.
result Spherical sectors are the only solutions under given conditions.
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 paper classifies and studies symplectic and contact properties of circular spherical divisors.
problem Investigating symplectic and contact topology of circular spherical divisors.
method Classification and analysis of concave circular spherical divisors, including embedding, Stein fillability, and rational homology type determination.
result All concave circular spherical divisors up to toric equivalence are realized as symplectic log Calabi-Yau pairs with minimal complements.
A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obt…
In this paper we study spherically symmetric monopoles, which are critical points for the Yang-Mills-Higgs functional over a disk in 3 dimensions, with prescribed degree and covariant constant at the boundary. This is a 3-dimensional gauge-theory generalization of the Ginzburg-Landau model in 2 dimensions.
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.
We present a boundary version of a theorem about solenoidal unit vector fields with minimum energy on a spherical domain of an odd dimensional Euclidean sphere.
We present in this paper a \boundary version" for theorems about minimality of volume and energy functionals on a spherical domain of threedimensional Euclidean sphere.
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.
New findings on hyperbolic groups and their boundaries.
problem Understanding the structure of cubulated hyperbolic groups with specific boundary conditions.
method Utilizing ideas from Markovic's work on Cannon's conjecture, focusing on quasi-convex subgroups and limit sets.
result Cubulated hyperbolic groups with certain boundary conditions are virtually fundamental groups of specific manifolds.
Suppose M1 and M2 are two closed (compact with no boundary) spherical CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of M1 and M2 also admits a spherical CR structure with positive CR Yamabe constant.
We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace--Beltrami operator uniquely determines the…
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.
Support Vector Data Description (SVDD) is a machine-learning technique used for single class classification and outlier detection. SVDD formulation with kernel function provides a flexible boundary around data. The value of kernel function parameters affects the nature of the data boundary. For example, it is observed …
New minimal annuli found in unit ball, solving old problems.
problem Constructing free boundary minimal annuli in unit ball.
method Symmetric and foliated by spherical curvature lines.
result First non-embedded free boundary minimal annuli in unit ball.
We prove that some Riemannian manifolds with boundary under an explicit integral pinching are spherical space forms. Precisely, we show that 3-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the L2-norm of their scalar curvature and…
We prove an Obata-type rigidity result for the spherical cap and apply it for an eigenvalue problem with mixed boundary condition.
The study proves a rigidity theorem for compact manifolds with boundary.
problem Rigidity of compact manifolds with boundary in low dimensions.
method Dimension reduction argument for mean curvature, extending Schoen-Yau's for scalar curvature.
result Sharp spherical radius rigidity and best NNSC fill-in in terms of mean curvature.
Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.
problem Characterize free boundary minimal submanifolds in geodesic balls of hyperbolic and spherical spaces.
method Define and analyze functionals related to critical metrics and spectral indices.
result Critical metrics of defined functionals arise from free boundary minimal immersions in geodesic balls of hyperbolic and spherical spaces.
A unique hyperbolic metric is found for each spherical cone-metric on the boundary of a hyperbolizable 3-manifold.
problem Finding a hyperbolic metric for a given spherical cone-metric on the boundary of a hyperbolizable 3-manifold.
method Constructing a strictly polyhedral hyperbolic metric on the 3-manifold such that the given spherical cone-metric is the induced dual metric on the boundary.
result The existence and uniqueness of a strictly polyhedral hyperbolic metric for a given spherical cone-metric on the boundary of a hyperbolizable 3-manifold.
Small bubbles sliding on a boundary maintain half-spherical shape.
problem Preserving the shape of small bubbles sliding on a boundary.
method Area-preserving Willmore flow, asymptotic analysis, convergence proof.
result The flow keeps a half-spherical shape for all times.
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.
Compact proof for Brown-York mass positivity and rigidity in flat and spherical spaces.
problem Positivity of Brown-York mass and rigidity of manifolds with mean-convex boundaries.
method Spinorial proof and optimal lower bound for eigenvalues.
result Optimal lower bound for first non-null eigenvalue of Dirac operator.
Classifies 3-manifolds with uniformly positive scalar curvature.
problem Classifying 3-manifolds with uniformly positive scalar curvature.
method Analyzes properties of 3-manifolds with mean convex boundaries and uniformly positive scalar curvature.
result 3-manifolds with uniformly positive scalar curvature are homeomorphic to sums of spherical 3-manifolds and S1imesS2. 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. 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.
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.
A spherical set is called convex if for every pair of its points there is at least one minimal geodesic segment that joins these points and lies in the set. We prove that for n >= 3 a complete locally-convex (topological) immersion of a connected (n-1)-manifold into the n-sphere is a surjection onto the boundary of a c…
In this new version, we give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in Cn,n≥2, is Kähler-Einstein if and only if the domain is biholomorphic to the ball. We establish versions of various …
In this note, we first give a criterion of pseudo-Einstein contact forms and then affirm the CR analogue of Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold of nonnegative pseudohermitian curvature on the space of smooth representatives of the first Kohn-Rossi cohomology group. Moreover, we …
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.
We show that canonical Carnot-Caratheodory spherical and horospherical metrics, which are defined on the boundary at infinity of every rank one symmetric space of non-compact type, are visual, i.e., they are bilipschitz equivalent with universal bilipschitz constants to the inverse exponent of Gromov products based in …
In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, pro…
Injectivity of geodesic ray transform on specific Finsler manifolds proven.
problem Injectivity of geodesic ray transform on spherically symmetric reversible Finsler manifolds.
method Reduction to invertibility of generalized Abel transforms using angular Fourier series and Taylor expansions of geodesics.
result Injectivity of geodesic ray transform proven on specified Finsler manifolds.
We describe the range of a restricted spherical mean transform, which sends a function supported inside a closed ball in a hyperbolic space to its mean values on the geodesics spheres centered at the boundary of the ball. The description resembles that of the same transform on the Euclidean spaces obtained by Mark Agra…
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…
We initiate the mathematical study of spherical collapse of self-gravitating charged scalar fields. The main result gives a complete characterization of the future boundary of spacetime, providing a starting point for studying the cosmic censorship conjectures. In general, the boundary includes two null components, one…
The so-called Mom-structures on hyperbolic cusped 3-manifolds without boundary were introduced by Gabai, Meyerhoff, and Milley, and used by them to identify the smallest closed hyperbolic manifold. In this work we extend the notion of a Mom-structure to include the case of 3-manifolds with non-empty boundary that does …
Study of immersions with Willmore energy leading to spherical and catenoid bubbles.
problem Classifying immersions with specific energy properties.
method Analyzing sequences of weak immersions with diverging conformal classes, applying Möbius transformations, and strong Wloc2,2-limits. result Obtaining spherical and catenoid bubbles as limits of immersions.